Lecture Notes in Mathematics A collection of informal reports and seminars

Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich

Series: Mathematisches Institut der Universitat Bonn Adviser: F. Hirzebruch

290

Don Bernard Zagier Mathematisches Institut der Un iversitat Bonn, Bonn/Deutschland

Equivariant Pontrjagin Classes and Applications to Orbit Spaces Applications of the G-signature Theorem to Transformation Groups, Symmetric Products and Number Theory

Springer-Verlag Berlin· Heidelberg· New York 1972

AMS Subject Classifications (1970): 57-02, 57B99, 57D20, 57E15, 57E25, 58GlO ------- ~

ISBN 3-540-06013-8 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06013-8 Springer-Verlag New York· Heidelberg· Berlin

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Offsetdruck: Julius Beltz, HemsbachlBergstr.

INTRODUCTION

This volume contains an assortment of results based on the Atiyah­

Singer index theorem and its corollaries (the Hirzebruch signature and

Riemann-Rooh theorems and the G-signature theorem). Because the

applioations of this theory have so wide a scope, the reader will find

himself involved with characteristic classes, finite group actions,

symmetrio produots of manifolds, and number theory of the naive sort.

On top of this, he may feel that the level of presentat10n is swinging

up and down in a dizzying fashion. I hope I may prevent, or at least

relieve .• his seasickness by a few preliminary remarks about the level

and oontent of the material.

The results ought to be comprehensible to a working topologist

(or even a good graduate student) who is not neoessarily a specialist

on the Atiyah-Singer theorem. The non-expert should thus not be put

off by references in the introduotion to esoteric theorems of Thorn,

Atiyah-Singer, and the like, nor be further discouraged when he finds

that even the first section of Chapter One throws no more light on

these matters. Background material is, in fact, inoluded, but it has

been postponed to the second section sa that the main theorems of the

chapter can be collected together at the beginning for reference. A

similar course has been pursued in Chapter Two.

Aside from this point, I should perhaps mention that a much

more tho~ough treatment of the required background on characteristic

classes, index theorem!! and group aotions can be found in the notes

[21] (if they ever appear), which also contain a further selection

of results in the same direction as those of this volume, and to some

extent complement it (overlap of results has been minimized).

IV

We now give a swnmary of the contents of the volume.

Hirzebruch defined for a differentiable manifold X a characterilltic

class

L(X) € HO(Xj(l)

which, on the one hand, is determined by the Pontr jagin class of X,

and, on the other, determines the signature of X. Thorn showed how

to define 1(X) when X is only a rational homology manifold.

Our goal in Chapter I wiLl be to generalize this to a defini tion

of an "equi variant 1-class"

L(g,X) I: H* (x;t) (g € G)

for a rational homology manifold X with an orientation-preserving

action of a finite grOup G. Apar t from their intrinsic interest,

these classes wilL make it pos~ible to compute the L-class (in Thorn's

sense) of certain rational honology manifol ds.

In the differentiable case, we define L(g,X) by

L(g,X) j,L'(g,X),

where L'(g,X) E H*(Xg;t) is the cohomology class appearing in the

G-signature theorem, j:Xg C X is the inclusion of the fixed-point

set, and j, is the C;ysin homomorphism. We then show (§3) that

(2)

(3)

1T*1(X/G) = l; L(g,X), (4) gEG

where rr:X ~ X/c; is the projection onto the quotient. Since the

map "*:H*(X/G;t) ~ H*(X;~) is injective, this completely calculate~

the L-class in Thorn' s ~ense ~r the simplest sort of rational homology

manifold, namely the Quotient of a manifold by a rini te group.

\Ve then imitate Milnor's reformulation of Thorn's definition to

gi ve a definition of L (g, X) for rational homology manifolds which

agrees with (3) for differentiable manifolds. Formula (4) still holds,

and indeed can be extended to c:alcull'lte t he new eouivariant L-classes

for orbit spaces.

As an example (§6), wa evaluate L(g,FnC) for g acting linearly

on Pn«:, and use this to calculate the L-class of pnt/G for G a finite,

linear action (the result had already been obtained by Batt). Also,

by studying the behaviour of the formula for L(g,Fnt), we can formu­

late various conjectures about the nature of the classes L (g, X).

The whole of Chapter II, whioh occupies half of the volume, is

an application of the result (4). We take X to be the nth Cartesian

product of a manifold M,and C; the symmetric group on n letters, acting

v

by permutation of the factors. The quotient X/G = M(n), called the

nth 5Y!l11Jletric product of M, is a rational homology manifold if

dim M = 25, and we can apply (4) to calculate its L-class. The complete

result is complicated, but displays a simple dependence on n, namely

LeM(n» j*(Qn+1.G), (5)

where j is the inclusion of M(n) in M(oo) and Q,G- E H·(M(oo» are

independent of n. Moreover, the "exponential" factor Q is very

simple and, so to speak, independent of M: we have

Q Qs(Q), (6)

where ry € H~(M(~» is a class defi ned canonically by the orientation

clas8 z € ~(M), and where

Qs(ry) :: 1 + 0-5 )1)2 + (5-s - 2'9-S ) 1/4 + ••• (7)

i5 a power llerie3 depending only on 8. The "constant" factor G,

though known, is very much more complicated,* and is only of any

real Ulle for manifolds with very simple homology. In ~13 , we compute

it in two cases: fOr M = 82s , where we find

Qa(1) - 1)QS(1)) Q (T)2 - 1)2

S

G

and for 3=1, i.e. M a Riemann surface. In the latter case, M(n) is

a smooth (indeed, complex) manifold and Qs (1)) = 1)/tanh T) is the

Hirzebruch power series. In this case LeM(n») was known (the Chern

class of M(n) was found by Macdonald), so we can check our main

theorem.

We Can restate (5) without the class G, in the form

j*L(M(n+1)

(here the first j denotes the inclusion of M(n) in M(n+1». ThiB is

reminiscent of the relationship between the L-classes of a manifold

A and submanifold B (namely j*L(A) " L(v)·L(B), where j:BCA and I'

is the normal bundle). A direct interpretation is impossible because

the inclUsion M(n) C M(n+1) does not have a "good" normal bundle,

even in Thorn's extended sense (this follows from (9) and the fact

that the power series Qs(t) does not split formally as a finite

(8)

(9)

• Since this volume was written, I have found a simpler expression

for G involving the (finitely many) multiplicative generators of

H·(M(~);~) rather than the additive basis described in §7. However, this will appear--if at all--elsewhere.

VI

product n~=1 (x / t anh x) for s >1). However, eq uation (9) seems t o sug~

ge8f . strongly t he e xistence of some more general type of bundl e

(possibly anal ogou s to the "homology cobordi sm bundles" defined by

Maunder .and Ma r ti n for the category of ;e - homol ogy manifolds) which

would be appropri ate to inclu ~ i o n5 of r a ti onal homology manifol ds

and which w ould pos c-ess L-cl asses. There is some reaSOn to be l ieve

t hat "line bundles" of thi s type would be classified by maps into t he

infinite symmetric product 525 (00). Since this space (by the theorem

of Dold and Tho rn ) is a K(Z,2~), such "line bundles" over X woul d be

classified by a "first Chern class" in (X, K(lZ ,2s ) ] _ H2s (X;lZ).

We e nd C hapter I I by calcul ating L(g,M (n)), where g is an

au tomorphism of a 2s-dimensional manifold M of finite or der p (then

g acts on M(n) via the diagona l action on Mn ). We find that (9) is

replaced by

j'L (g, M(n+p ) )

if P is odd, and ha s no analogue at all i f p i s even. Again we

have the possibil i ty of checking our result5 in the two-dimensional

Case, this time by taking M= S2 and comparing w i t h the results of

Chapte r I on the Bot t a ction on Pnt = S2 ( n).

(1 0)

In Chapte r III we make explicit ca lcul a t ions with the G- s ignature

theOrem on certa in simple manifolds (FnC with t he Bott action,

Brieskorn varieties, and related manifolds), a nd relate them to the

number-theoret ic properties of f'ini te trigonometric sums such as

p- 1 11;". 11;n~" (_1)n E co~ .•• cot~

j= 1 p P

(where p ~ 1, Q1' ••• ,Q2n integers prime to pl. We prove that (11) is

a rational number whose denominator divide s the denomi.nator of the

Hirzebruch L-poly nomial Ln (i. e . 3 for n= 1, 45 for n=2, etc. ) . We

also prove a new'~eciprocity law'for the expre ssions ( 11), both by

elementary methods and--in two different ways--by specializing the

G-signature the orem.

AI though i t is not lIIRde a.pparent here, there is a close tie

between the 1'e suI ts of Chapter III and the result i n Chapter I on

the 1-c1a55 of Pn¢/ G (cf. [ 21]) •

• .. ..

(11 )

VII

The re3earch described in this volume took place in Oxford and

Bonn during the years 1970-71; I would like to thank both of these

institutions, as well as the National Science Foundation and the

Sonderforschungsbereich Theoretische Mathematik del" Universi~~t Bonn

for financial support. Above all, my thanks go to Professor

Hirzebruch, who taught me the little I know and much mOre •

• • ...

Notation is fairly standard, except that for want of italics

we have underlined symbols occurring in the text (not, however,

Greek or capital lettere Or expressions containing more than one

letter: thus we would write "let,!!; be a point of a 3et A" but "then 2rix )

iI. equa13 e .". We use IAI to denote the number of elementll

of a finite set A.

References to the bibliography have been made in the normal

way, by the ulle of appropriate numbers in souare brackets; an

exception is the reference Spanier [38] which like everyone e13e

we refer to simply as "Spanier. IT

The numbering of theorems, propositions, lemmata and equations

starts afresh in each section. The symbol §3(10) denotes

equation (10) of aection 3.

TABLE OF CONTENTS

CHAPTER I: L-CLASSES OF RATIONAL HOMOLOGY MANIFOLDS ••••••••.••••••••• 1

§1. Summary of re:sults ••••••••••••••••••••••••••••••••••••••.•••••• 3

§2. Preparatory material ••••.•••.•••..••••••••••••••••••••••••••••• 6

(I) Homological properties of manifolds ••.•••.••••••••••••••• 6 (II) Milnor's definition of the L-class of a rational

homology manifold •••..•••.••.••.•••••.•.•.••.•.•••.•••••• 8

(III) The G-signature theorem ••••••••.•••••••••••••••••••••••• 10

§3. Proof of the formula for L(X/C) ............................... 14-

§4. A definition of L(g,X) for rational homology manifold:s ••.••••• 20

§5. The formula for Leh',X/C) ..................................... 23

§6. Application to a formula of Bott and some remarks on L(g,X) ••• 25

CHAPTER II: 'L-CLASSES OF SYMMETRIC PRODUCTS ......................... 32

§7. The rational cohomology of X(n) ............................... 34

08. Statement and discussion of the formula for L(X(n») •..••.••••• 40

TABLE: THE FUNCTION Qn (t) AND RELATED Fo-.\r~R SERIES ............ 51

')/9. The action of Sn on Xn ........................................ 52

910. The Gysin homomorphism of the diagonal map •..••.••.•••.••••.•• )9

§11. Preliminary forllJUla for L(X (n)) ............................... 63

§12. The dependence of L(X(n) on n ................................ 67

§13. Symmetric products of spheres and of Riemann surfaces •••..• , •• 71

§14. The equivariant case .......................................... 77

§15. Equivariant L-classes for sYlllIlletriu products of spheres ....... 91

CHAPTER III: THE G-SIGNATURE THEOREM AND SOME ELEMEl'JTARY

NUMBER THEORy •••••••••••••••••••••••••••••••••••.•••••••••••••••••••• 96

§16. Elementary properties of cotangent sums ••.•.••.••...•.•••.••• 100

§17. Group actions and Rademacher reciprocity •.•••••••.••••.••.••• 110

(1) A group action on projective space ••••••••••••••••••••• 110

(II) A group action on a hYPersurface ....................... 113

§18. Equivariant signature of Brieskorn varieties ................. 118

REFERENCES ••••••••••••.•••••••••.••.••..•••••.••••••.••.•••••.•••••• 128

CHAPTER I: L-CLASSES OF RATIONAL HOMOLOGy MANIFOLDS

In his :famous paper "1ee classes caracteristiques de Pontrjagin

des varietss triangulees" ([l~J), R. Thom showed that it is possible to

define a Hirzebruch 1-cla55 L(X) € H*(X;~) (or equivalently a rational

Fontrjagin class) :for a rational homology manifold X, in such a way as

to obtain the usual L-cla5~ if X possesses the structure of a differen­

tiable manifold. This definition rested on the possibility of making

precise the notion of a rational homology submanifold of X with a

normal bundle in X, and showing that X has enough such submanifold! to

represent all of its rational homology. The definition was later

simplified by Milnor [:51J, who observed that it is easy to give a

definition of • "submanifold with trivi .. l normal bundle" agreeing with

the usual concept i:f X is differentiable (such a manifold is f-1(p),

where f is a map from X to a sphere and £ is a pOint of the sphere in

general position), and that it follows from the work of Serre ~~ that

there are also enough of these more speCial 5ub~nifolds to represent

all of ~(X;~) (indeed there are just enough, i.e. ~ one-one correspon­

dence; in Thorn's definition each homology class w~s represented by many

l!ubmanifolds and one had to check consistency ~s well ~s suffiCiency).

Nevertheless, the definition rem~ined essentially an exi~tence proof

rather than a procedure for actually computing 1(X), and as a result

the definition hal! remained of relatively little intrinsic interest

and has been most important for its use in proving facts about the

ordinary 1-cla55 or rational Pontrjagin class (e.g. that this is the

IIams for two differentiable manifolds with the same underlying PL

IItructure) .

There is, however, one especially 5imple type of rational homology

m,mifold, namely Ii quotient space X/G of a smooth manifold X by an

orientation-preserving ~ction of a finite group G, and for such a sp~ce

it is possible to give a formula for the L-class in terms pf the action

of G on X by using the G-signature theorem of Atiyah and Singer. This

formula will be given in §1 ~nd proved in §3. An illustration of it

will be given in §6, where we calCUlate LeX/G) for X = Fnt and G ~

- 2 -

product of' f'ini te cyclic groupe acting linearly on X; the 1-cl&1I11 of

thill IIpace had already been calculated by Bott using a different method.

A much more difficult application ill to the L-cla~8 of the nth 5ymmetric

product M(n) of a manifold M (here X = Mn and ~ is the symmetric group

on ~ letters, acting on X by permutation of the factors); thill will be

carried out in Chapter II.

In the formula for L(X/G), certain cohomology classes L(g,X) ~

H*(X;~) occur, defined for each gEG and such that L(id,X) = L(X).

Their definition in the differentiable case is based on the G-signature

theorem and thus reauires a knowledge of certain normal bundles and of

the action of ~ on these bundle8, 50 that it depends very heavily on

the differentiable structure. However, it is possible to define these

"egui variant L-classe:s" also when X is only a ratiomll homology G-manifold

in a maImer exactly parallel to Milnor's definition in the non-equi variant

case. This definition will be given in §4; we then show in §5 that the

formuh. obtained for 1(X/c;.) in the differentiable Case holds more

generally when X is a rational homology G-manifold, and indeed can be

generalised to a formula for L(h',X/G) where h' belongs to a finite

group of automorphisms of X/ G induced by automorphisms of X.

A more precise statement of the results proved is given in §1.

The following conventions will apply throughout: the word

"manifold" will alwaYl!! refer to a connected, close'l ( ,. compact and

without boundary) manifold, differentiable unless preceded by the

words "rational homology." The coefficients for homology and cohomology

will always be one of the fields ~, ~, or ~ of characteristio zero or

else a twi8ted coefficient system locally isomorphic to one of these;

thul!! there will never be any torsion. We will omit notations for the

coefficient homomorphismll, so that, for example, we will multiply the

class L(X) E H*(XjQ) with element8 of H*(X;t) without explicit comment.

Cup products will usually be denoted by juxtaposition but sometimes

written out as xUy. Evaluation of a cohomology class on a homology

class will be indicated by juxtapollition or by < , >. A class in the

cohomology of a disconnected fixed-point set Xg is a 01a55 in the

cohomology of each component, and expressions like L'(g,X)rXB] are to

be interpreted as sums over the connectedness component8 of the

corresponding cohomology classes evaluated on the fundamental class

of the component in question.

- 3 -

§1. Summary ef re~ulte

Let X be an orieated clo~ed manifold Gn which a finite group G Qcta

by orientation-pre~erving diffe.mcrphi~ms. It is known that the ~ignature

.f the qu~tient space X/G is the average over G of the equivariant

~ignatures Sign(g,X) (of which a precise definition will be given in §2). The complex numbers Sign(g,X) can in turn be calculated frram the

G-signa.ture theGrem I!)f Atiyah lind Singer, which 5h.tel'J that

Sign(g,X) '" <L'(g,X),[xg]>, (1)

where Xg is the submanifold of X consisting of points left fixed by S and L'(g,X) .. H*(Xgj (1;) il! Ii certain eohomolvgy elalll!;, explicitly given

in terms or the characteristic classes of Xg and its equivariant nQrm.l

bundle in X. If Xg ill JII~t Itrientable, then b5th L' (g, X) and the

fundamental class are to be under~toQd with the appropriate twisted

coefficients; thill will be made more preciM in §2.

We ;;;11;0 know that X/G il; a rational hom.logy manifold, and therefore,

by the work of Thorn and Milner, h;;;:5 II rational Pontrjagin ch.~!5 and Ii.

rational 1-cla55. Since this cla55 ill det~rmined by the signaturel5 of

the various submanifolds (or rather ratienal homology svlmtanifslds),

it is rea50nabl!l to assume that the L-clas!5 .f X/G caJ!! be c .. lculated

alS was the signature, namely by averaging over G some equivariant L-clB.ISS

in H* (xl. To get fnm the cohomology of X/G to that of X we simply

need to apply n*, where n i~ the projection map from X t. X/G. We

therefore can reasonably expect a rormula or the form

1T*1(X/G) 1

TGT L: L(g,X) gE G

h held, where L(g,X) :LIl a cla:;s in H*(xl defined lSohly by the action

of is. em X. For the definition of L(g,X), we cbl5erve that the G-signature

theorem already prllvide5 UIS with a elaM in H* (Xg ). The map j'" induced

by the inclusion 1 of Xg in X goes the wrong way to define L(g,X) from

L'(g,X), /5(1 we use intltead the "Umkehr homemorphism" or Gysin homo­

morphil5m j!: H* (xg ) ..,. H* (X). defined by pa~~ing to homolGgy via Poincare

duality and then applying j .. in hemD10gy (we will define all of thel5e

concepts mere preci~ely in §2). Finally, tD have (1) hold, we ~eed

to inzert a factor (deg N) to c&mpen~ate for the aifference between

- 4 -

the olaMes "*[X] and [X/G] in f4(X/G). Thus the flll"lllula. whioh we would

expeot te hold, and which will be preyed in §3, is:

Theorem 1: Let G be a finite greup. and X an erientable, clei'!sd,

differentiable G-manifold. Let 1I:X-+ X/G denete the proj"ctiol'l map,

j:Xg C X the inclusi,m of the fhed-point set of an element gE G,

and L'(g,X) E H*(Xg ) the Atiyah-Singer class. Th,,~

deg 1T 1I*L(X/G) 1

m Z L(g,X), g E G

where

L(g,X) j! L'(g,X) € H*(X).

(2)

(3)

Ne make Ii few comment~ ab0ut the statement of the theorem. The

class l' (g, X), as stated above, ffiQy lie in H* (Xg; ~) with ~ Ii twisted

coefficient system locally isomorphic to ~ (~ is the tensor product of

the orientation bundle sf Xg with the trivial bundle with fibre ~), but

i'rllm the definition of j, it follows that the claM L(g,X) defined by (3)

is an untwisted class, so that the summands 0n the right-hand side of (2)

arl'! elemenh .. f H*(X;~). Ne th"n deduce f'r.u (2) th .. t the sum liee in

(the image in H*(X;~) 9f) H*(X;~). Thu~, even if we are not interested

in the quoti"nt X/e- or its L-clas8 ,,1'1 such. 1'0"11' stHl get jnteresting

information about the G-space X itself from equation (2), namely a sort

of integrality theorem for the cohemelegy claeses defined by (3). Th"

Atiyah-Singer result enly gives the top-dimen5ional component of this

(i.e. the eignatures), but thon gives a stronger result: the average

over G- ~f the complex (algebraic) number5 Sign(g,X) is not only a

rati@nal number, but even a rational integer.

The next point about formula (2) is that if G acts effectively

(which c an always be assumed by factoring ~>ut the normal subgroup

which acts trivially), then deg 1I = IGI, 50 the numerical csefficients

Can be omitted fr~m (2), simplifying it somewhat.

Finally, it is known that, for cohomology with rational or complex

coefficienti'! (or, more generally, ~ny field ~f characteristic zero or

prime to Ie-I fi8 coefficient~) the map 1T* induce:s an isomorphism

rr*: H*(X/G) __ H*eX)G C H*(X}

frem H*(X/G) onts the G-invariant part of H*(X) (Grothendieck [1~,

- 5 -

Berel [1]). In particular rr" is injective, 50 eg. (2) determines

the L-cla~~ of X/G completely.

Th""orern 1 as st2ted above is the fir~t main result ef this chapter.

However, its proof suggests the p$ssibility of defining the cohemology

classes L (g, X) e H* (X; a;) when X is jll~,t an oriented r:.t tiQmll hemolegy

G-manifold . Here we cannet use a fermula such as (3), since it. i~ net

pe55ible to define the Atiyah-Slnger class fDr nen-differentiable

Actions, Bnd indeed it is not clear that the class L(g,X) that we

define vanishes if £ acts freely "n X. But we C/iln sti.ll define the

class, in a way exactly parallel t. the Milnor definition of L(X) fer

a rational bernolegy manifold X. This will be done in §4. Once L(g,X)

is !iefined, eq. (2) makell lIense ev~n for ratjonal hornaltlgy lIl&nifl!llds X

(since xjG is t hen aha a rational hemolGgy manifold and has an L-clas5),

and we prove in §S that it still holds. I~deed, we can generalise it:

The~rem 2: Let X be an oriented rati onal hom.l~gy G-manif(}ld. Fgr each

£ in G, let L(g,X) in H*(X;~) be the cla~3 defined in §4. If h i~ an

automorphism of X of finite order which oommutes with the action of

G OR X, and h' the induced autmmnrphi sm .f X/C, we have the relation

de~ 11 rr"L(h',X/C) 1

TaT E L(gh,X). geG

Finally, in §6 we evaluate explicitly the quanti t i <)s L(g,X) for

X = complex projeotive space And G aoting li~early en X. Thi~

calculation will be used in §15 to check a general formula for L(g,X)

when X is the nth symmetric product of a manifold (the space Pn~ is

the nth lIyrrunetric product of S2).Nhen we put the value er L(g,X)

inte Theorem 1 we obtain a. formula fer L(X/G) al ready obtained by

Bott by "ther methodl5 (unpublished; :!lee, however, Hirzebrllch [1('J).

- IS -

§2. Preparatory material

Thi~ ~ection c.ntain~ more detailed de5cription~ of 50me of the

concepts and thellrems which were used i n §1 for the fnmulation of the

various re~ults stated there. Ne do not discuss definitions or re~ults

which are very welI knawn. Thus, for example, 'lie assume the definitions

or the :signature and the L-class .f a manifold and a knowledge of the

Hirgebruch index theorem, but define the eguivariant versions or these

notions and state explicitly the G-signature theorem of Atiyah and

Singer. We also define rati onel h8malogy ~Rnifolds and give in some

detail. Milnor's formulatilln of the definition of the L-clas5 for a

rational homology manifold (that a definition i~ ~ot5ible had been shown

by Them). This will be especi ally important t. us s ince we will copy

the constructi on in \?4 for the definition of the eruivariant L-cla5s

L(g, X) for raticmal homology m"nifolds X. The only poj.nt we need t.

make for 8 reader acquainted with these ideas and wishing t<!> skip this

~ectiQn is that the class L '(g,X) appe crine in §1 i~ not exactly the

cohomeltlgy 01a5s appearing in t he original f ormulation of the G-sign"ture

theorem CAtiyah and Si nger [1)) but differs from it by a power of twe

in each dimension (except the oemp~nent of tep deGree, which for b.th

classes is equal to Sign(g,X)).

We break up the 5ecti &n into three parts, In (I) we discuss

variou:s he molillgical propertie s of manifold5; the definition Itf 8

r.tienal hOIlle> l ogy manifold, the orienta tiC)n system or local coefficient!!

rer a mc,nif.ld, and related concepts (the Thorn clns5 of Ii nGln-()riented

bundle, Poincare du! Ii ty fer a non-Ilrientable manifold, the Gysin

homomorphism). A descriptien af Milnor's definition of the L-cla5s of

" rati .mal homology manifold t hen follows in ( II), while (III) contain5

the definition of Sign(£,X) for a G-manif old X and a s ta teme n t of the

Atiyah-Singer G-si gnature theorem.

(1) Homologlcel properti8s !£ manifolds

A ratienal homology manifold af dill1ension !! is a triangulated s p;;ce

in whioh the boundary (If the ll t ar of each vertex has the same ratiQnal

ham8logy grlllups as Sn-1 Equivalently, it is a simpl iciRl complex X

such that

- 7 -

ror .. 11 l( £ X. Then we can define II system It[ 1$ca1 coefficients for X,

denoted orX and called the orientation system of X, which at the peint !

is just the vector space ~(x,X-!~;~) (where we do not choose a specifio

isomorphl.sm with ~). There is then an IIrientati8n ~

(2)

whose imase in H. (X,X - [xl) for :z: E X is the identity in

(3)

Notice that this is independent of the particular isomorphis:ns of

H~(X,X-lxl) with ~ given by (1). If the system orx of local coefficients

iB trivial, X is said to be orienteble; then the element (2) is in Hn(X;~),

If r is any system of local coefficients f6r X, then the cap

product with [X] gives a Poincare duality isomorphism:

n [X]

If f:X~ Y i~ a map between twQ manifolds, the Gysin homomorphism 1'1 is

wh!'lre r is Ii. local ccefficient system over Y, f*r th!'l induoed system

over Xt and 1'* the map from H,.(X;f*r) to H,.(Yjr). We will only need

this when f. is an inclusil!ln map bet\'leen differentiabl!'l manifnlilll Qnd r",~.

If e is a real veotor bundle ef dimension S over X, it al~e

defin!'lB an orientation system er local coefficients, which at X€X is

the fibre of e at ~ being den.ted Ex' Tlu5 defines a ~ class

E

(6)

where E. ill the total space E 01' E minus the zerfl-section X, Tf is the

projection map E ~ X, and rr* or c is the system ef local c(lefficientll on E s

induced from ore by rr. Namely, the Thom class i~ defined uniquely by

the reqUirement that its restriction to any fibre ill th!'l identity of

Hq(E E "rr*or) ~ Hom(R (E ,E ), H (E ,E )). x' ex' e q x IIX q X ox ((J )

- tl -

Then the Euler class .f ~ is defined as the relltriction te the zero­

section X of the Thorn clas~; thu~

(9 )

The bundle e is oriented if or~ is trivial; then its Euler class liell

ill. H* (X;:E) (or II" (X; ~) if vre used rational coefficients in (6)). This

is th!! case if and only if the fir~t Stiefel-'Nhitney C181111 of e ill zero.

In general, the first Stiefel-iVhi tney claM determinell the orientation

",ystem of c<gefficients; thus when we take a direct "lum eQ} 1) of bundlell,

the Ol?ientatien lIystem of the sum is the tensor product or eellr1) , while

the Stiefel-',Vhitney class of the sum is the sum w1 (e) + w1 (1)). In

particular, since Stiefel-Nhi tney classes have order two, it deell not

matter in equations like (4) whether we tensor with orX or erx- i (we

have alre'ldy used thi5 freedom in eq. (5), where th", positionll of

r 8nd r ® orX h"ve be",n interchanged). ~'il'!ally, if X is a differentiable

equals orTX' where TX is t~ tang"'nt bundle.

(n) Milner'l\ defini tilln Itf the L-ol&s6 of a rational hamelg1>Y manifold

It has been preved by Serre [.7] that, if X is a eN cemplex of

dirnen:si-:m n" 2i-2, th..,n hometepy classe$ gf map~

f: X ... ( 10)

form 6ln aheHan greup ui(X) (the i th cohomehpy grQup af X) which,

up to torsiun, is isemerphic to Hi(X). More precisely, the natural

map (X) -7 H:1 (X) sending the map (10) to f*u (l'Ihere u E H\Si) ill

the generator) become5 an isomorphillm after ten5ering l'Iith (J. ThUll,

for any element x e ~(x), some multiple Nx can be written as r*u for

some f:X ~ Si whose homotopy type i5 unique up to torsion in !TieX). Now let X be Rn orient",d ratiltnal h~mology manif~ld af dimeullion ~.

Then ther", iB a fundamental 01,,55 [X) in HA(X;~), and therefore we

have an int",rseotisn form &n Hk(X) (if n=2k) defined as usual by

sending tWit element" t. the evaluatiltn of their cup preduct on [X];

thus the signature of X is defined. New one can prove th", foll.wing

facts: if f h; a sirnplicilil map as in (10), where 3 i has some fixed

standard triangulatien, then the inverse image Gf a peint,

( 11 )

- 9 -

i~ ~n &riented rati&nal homol&gy submanirold or x Cor dimensien _-i)

r or almo stall p E Si , alld moreover, the c"b ordism cl ass or A--and

hence also its signature--are independent or the point ~, again ror

almost all £. This define/', then a number I(f) = Sign(A), which only

depends on the hgm~topy type or A. Moreover, from the definition of

the addithn in vi(X) '>ne ea:lily finds I(f1+f2 ) .. I(f1) + I(f2 ),

:10 the map [f] ... Ief) defines Ii hamomerphism

If we teneor this with ~ and cembine it with the theorem or Serro

stated above and the Poincare du"li ty i:lemorphi:lffi in X, we obtain

a mrique olasl!

1 . € Hn-i(X;(Q) n-1

I!uch that

Sign(A)

rer 0i11 map/', £. as in (10). Thill ~ll only holds fer n';; 2i-2 sr

n-i ,;; (n-2)/2, but we COin define 1. als8 for larger ..i. by oh(ua.sing J . N

a large integer N, defining the class 1 ~ E HJ(XxS) by the ab9ve '1 J

( 12)

( 14)

procedure applied ts XxSl\ .md then lettin6 1. be the c&rre",pondin6

class under the isomorphism af Hj(X) with Hj(ixsN). Then the L-clas5

of' X i~ defined a5 the lIum of' thl'!~e clall5es:

L (X) 00

6 1 j ~ fl* (X; ~ ) • j",O

It i~ easy te see that thi5 agrees with the usual definition if

X is ,. differ"ntinble m.mifeld. Indeed, since the 1-c1a55 fijI' II preduct is

multiplicative and L(SN)= 1, we have L(X) = L(XxSN) (Where we have

identified the cahomologie:s of the two space~ in dimen$i.sns up to n «N),

50 we ol'lly have til check that the usual L-clai'lIl satisfies (14). But

the map! oan be chosen within itl! homotopy clas:! all difft;rentiable; t:1en

A ill a differentiable submanifeld er X for almellt all 2 in Si and hal!

trivial normal bundle (llince a point has trivial nermal bundlo), :so the

1-chll5 ttl' A is the relltriction j*1(X) of the 1-claM .r X (where j:ACX).

Also, f*G E ~(X) is the POinoare dual of j.[A] e Mn_i(X). Therefore

<L(x)Vr*a,[XJ> =<L(X),j.[A]> = <j*L(X),[.A]' = <L(A),(A]> '"' Sigl!(A),

- 10 -

so the L-clas~ in the u~ual seneo satisfies (14). Since equation (14)

defined uniquely the ala~s lj (fer X a ratianal hemology manifold),

we ,;ee th"t th", clal'is (15) is ind""d the mllial LeX) if X h differentiable.

(III) The G-sign.ture theerem

If • greup G .ats on an orientable rational homelogy manifold X

(here and in future WI! will tacitly aMume that all grouplI are compact

Lie greup~ and that actions preserve any ~truature pre~ent; thue the

action here must be simplicial and preserve the orientation), there is

defined a complex number Sign(g,X), depending osmly on the action af g*

on H*(X;~), for every gE G. The definition ie .e f.ll.w~: If X ha~

odd dimension, we ,;et Sign(g,X) = O. If X has dimension 4k, the

interllection rorm B(x,y) = <xU y, [XJ > ill eymmetric, non-degenerate

(becaul'le .f Paincare duality), l'.nd G-invariant (eince g*[X] = [X]

by ;;.ssumptian). Ne coan .there:fu·e decampase the middle cehernalogy . .2k _ \" a G-,~~a.<,a'" w3.y _ • •

greup H (X;~)/a.s II d.l.rect !\um H ill H , IIIhere the bl.llnear farm B + -

i!l pa!litive definite on H+ and negative definite an It. Than

Sign(g,X) tr(g*IH) - tr(g*IHJ; +

(16)

this definitien is independent ar the decampasition ~~)= H ill H . + -

If X has dimen!liQR 4k+2, then B(x,y) is skew-!lymmetric, nen-degenerate

~nd G-invariant. Then if we choase R G-invariant pmsitive definite

inner prllduot < ,> en H2k+1 (X;IR) and define an aperatar A by requiring ( ) 2k+ 1 ( ) . B x,y = <Ax.y> for .. 11 x,y€ H X;IR , the eperater A le :skelf-

adjeint, !Ie the eperatsr J = A/(AA*)1/2 has square -1. This gives 2k+1 ( ) H X;lR the lItruoture .r Ii camplex vl!Jcter 1Ipaoe, and since gOO

cemmutes with J, it ach on ~k+1 (X;~) in a complex lineliT wlIy. Then

Sign(e,X)

where the trace is taken @~ the map g* th.ught of as an aut.rn.rphieffi

af a c~mplex vector ,;pace. Again this i1l independent ar the cheices

made. Thi" completl!J!\ the definition of Sign(g,X) in all cases. If

g1 acts an Xi and g2 acts Qn X2, then the signature .f the product

actien ar g1xg2 an X1xX2 is give~ by

( 17)

(18 )

The centent of the G-signature the.rem is a farmula rar Sign(g,X)

- 11 -

i. terms ef the fixed-ptint set Xg and it~ equivariant normal bundle,

in the C.1ll'le that X is .!l differentiable manifold. Tt state this ftrmula,

we first define oertain char~cteristio Clas8es ef ctmple~ and real

bundles (i.e. multiplicative sequences in the Chern or Pentrjagin

olaasea, respectively). If e is a real number, not a multiple tf ~.

and e is a ctmplex bundle of (complex) dimension q over a space Y,

we define

n j

where the Xj have the u!'\ual interpretatitn as fermal tw~-dimen!'\ional

clhtmelogy classes suoh that the Chern Cla.M If e i:>

c(e) n(1+x.). j J

Thu!l Le(e) is an element ef the $ubring H*(Y;~)[ei8J of W(Y;(:).

If; i:l it real bundle ever Y, 1'I"e let

L(e) E H*(Y;~)

be the Hirzebruch L-c1al'l:l .f e , defined R!'I

is the Ptntrjagin class tf';. We let

n X,j tanh Xj

be the Euler class (cf. eg. (9) "bove), Rnd define

",here

( 19)

(20)

(21)

(22)

which :h legitimate since L(e) hu lead.ing c.efficient and is theref'ere

invertible, Netice that if e is a cemplex bundle and we set U=H in (19)

(where lHI have chtsen the number .f x. 'st. be equal tt .9,) we ebtalJl, J

after first cancelling the (zert!) facter1!) ooth i6/2 frem aumerater

and deneminater, a preduo" et the tanh x.. This then agree~ with (23), J

sinee eCe) '" c (e) '" X1" ,J[ and Lee) :: n (x./tll.nh x.) in this case. . qq. J J We return tt the aetien tf .s. tn a differentiable manifeld X. The

fixed-peint aet Xg is a sm •• th ~ubm.nittld, net necessarily trientBble;

we dfmet" i til lJIerllllll bundle in X by Ng . At each peint ! er X, the

aetien et .s. en the fibre Ng CaA be decempesed, by stand8rd representatien x

- 12 -

theery, a~ Ii. ~urn af one-dime~ianal eub~paces en which ~ act~ as

multiplication by -1 and two-dimeneienal e ub ep~cee an which £ act~ by

ca~ e -sin CI ) sin 11 cas CI '

(24-)

where e is a real nu mber net divieible by rr (the eigenvalue +1 can.et

eccur en the nermal bundle H€;). Sj nee Ae and A -e are equivalent, ."e

can a~lIum. that 00; eo; Tf rer the repres.Atatien~ (24), and write

Nt; Ng Iil l: Nt; (25 ) .ll X, 11

0<0<11 x,e

where Ng X,l1

h the subspa.ce en "hich if. act~ a~ -1. Each af the spaces

Ng (00;8<rr) has a natural cemplex IItructure en whi ch ~ ach as x, e ie

multiplicatian by e T he decompasition (25) af a fibre extends t-

a decamposi tian or the whale bundle ~ as

Ql 1: (26) Q<b<l1

where N; i s naw a real bundle over Xg en whi ch if. acts as -1 and N~

" 1 b d Xg h" h i8 I 1 ~s a cemp ex un le ever an 'II lC if. acts as e n particu aI',

the bundle~ N~ all acquire a natural arientatien fram the camplex

structure, SiAce X aleo has a given arientatian, we abtAin frem (26)

(ud the relati en j'" (TX) " NG (jI TXg ) an isemuphi:lm betweiln the

~y:ltems .f t~i~ted ceeffici~nt~

Ws .aw defjne Q cla~~

L'(g,X) E

where l'Ie have used the chllract eri:ltic clas:lell defined above and the

is.marphism arXg ~ arN~ given by the pre~cribed .rientatien~ en X

and en the cl!lmplell. bundlM N~. Finally, the fUl'ldament.l cla:ls [Xg ]

alsQ lies in the hamelegy gr.up with caefficient:l erXg ' 5e we can

evalua.te (27) on thi~ cla:!!:!. The AtiYlih-Singer G-s ignature theerem

!ltatei5 tha.t thi!l is preCisely Sign (g,X):

Sign(g,X)

(27)

(28)

Saurces: The material in (I) i3 3tandard; it can be raund sketchily in

Spanier [38J er Ddd [8J and in detail in Heithecker [11 J. The eriginal

- 13 -

paper et Them preving that L-clas!e! can be defined fer ratienal hemelegy

mllnireld~ in II way c.nl'lil'ltent with the previeul'l defini tiel!. in the

differentiable cal'\e is Them [~~. The definitien ef rational hemelegy

manifeldl'\ as given in (I) and t~ whele cententl'l er (II) are taken fr_m

Milner [~1]. Finally, the material in (III) c_mea f'rem Atiyah and

Singer (1).

- 14 -

The ~tarting point fgr the pro.f .f The.rem 1 ef §1 will be the

cerre~pending relation fer signatures, namely

Sign(A!G) 1 ill 1: Sign(g,A) gEG

rer a G-manU.ld A (G finite). In fact thb also hald.$ fer Ii rational

homolegy manifold A, ~ince it~ pr.ef depend~ enly en the definition

of Sign(g,A) in terms er th. actien er g* en H*(A). It is trivial if

A has edd dime n~i en (b eth si de ~ are zer.) or dimensi on 4k+2 (then the

left-hand side is identically zere, while Sign(e,A) = - Sign(g-1,A»).

If the dimension or A is 4k, we use the fact (cf. §1) that

(2 )

where rr denotes the pr0jectien map frem A t. A/G (this held~ becau~e

we are always working with a fiela ef oharacteri5tic zere as ceefficientE).

Then cemparing the definitien er Sign(g,A) (§2, (16» in this case

with the definition ef Sign(Jv'G). we find that (1) reduces te the

elementary identity er linear algebra

1 ill I: tr (gIV)

gE G

rer the G-invariant part ef Ii. vecter space V. ',Ve will apply this tit the ~et A = f-1 (p) appearing in Milner r s

definitien ef L-ola~se~ (§Z, (11)). Since we are intere~ted in the

L-cl~~~ _f X/G, we begin by ch~.sing a ~implicial map

We then define an equivariant map! frem X te Si by

f - i fo1T: X ..... 3

i Gle arly, ! i~ G-equi variant with G acting tri vi.lilly en S , i. e. it

i~ G-invs.riant

(5)

(6)

3.Proof of the formula for L(X/G)

- 15 -

Thull A ill S G-invariant lIubllpace ef X.

We call. change f within it~ equivariant hemetepy cla~s (i.e. threugh

map~ factoring threugh X/G) te a differentiable map. Indeed, we juat

replace the cempe~itieJI. f: X"* Si-> lRi+1 by a diff'<trtul.tiable map

f':X-> Ri+1 with max If(x)- f'{x)1 < e, Utd then define Ii third map

f'" frem X te Ri+1 by

fll(x) 1 TGT ~ i" (gox) •

g e G (8)

Thl!ln fll i:l G-equivarisAt rand difft!lrentiable alld, if e is small eneugh,

clese te f; in particular f"(X) C lRi+1 -!Ol ~e we can cempe:!e with

the prejeotien lRi+1 -{Ol'" Si te ebtain a map f~' :X'" Sl which is

G-equivariant, differentiable, and arbitrarily cle~e te f. If we

apply the whele precess te f t (Where f t i3 a hemetepy trem f te fl,

e.g. the linear ene) we ebtain al!. equivariant hemetepy frem f te riff • We theretere as~ume that f i~ differentiable. Fer the whele et

Milnor'~ definitien we are allewed t. exclude ~et:l ef mea:lura zere

fer the paint ~ in Si; thu~ here we can use Sardt~ theerem to have

~ a rt!lgular value .r the function f and af each er the (finitely

many) functions f I Xg (g € G; natice thilt Xg ill a differentiable

manifald). Then A i5 a differentiable submanifeld ef X and meets

each ~ubmanif'.ld Xg tran$ver$slly with inter~.ctien Ag.

',Ve new wri ttl! L fer L (X/G) fer cenvenisnoe; the. Milner t ~

definitien 5ay~ that L is uniquely determined by

Sign(A)

(again fer almut all p"-Sij ws are al~e ignering the preblem ef

defining the clas~e5 lj for l larger than half the dim&n~ian af X

~ince we can remove such dimen~i~nal re~trictiens by multiplying

everything with a sphere of large dimension on which G acts trivially).

We new apply (1) te expre ~s the right-hand side ef (9) ill term~

ef the quantitie" SigR(g,A), and th",n the G-signature thearem (§2 (28)

t. evaluate the latter. ThUS

(Lvr*a)[X/G] Sign(A/G) 1

m ~ Sign(g,A) gE G

- 16 -

1 m r: L'(g,A)[ Ag ]. g€ G

The G-signature the&rem i1l applicable because A is "- differentiable

ml3nifdd en wh ich G II cts.

In the d.iagrem

A __________ t~ ____ -+. X

(10)

.f differentiable manif~lds and inclu1Ii~n rnQPs, the inclu 51~n5 i and i'

censist ef the inclu:5ion ef the inver1le i mage Clf a peint (ef the

differentiable maps! and fiXe, respectively) and t herefore have

trivial nermal bundle. It renews th"t

where N(Ag ) denetes the nermal bundle ef Ag in A, N(Xg ) denote1l that

.f Xi; in X, and e denetes " trivial bundle (of dimension i). This

ieeroerphisffi i ~ even G-equivariant since A = f-1(p) and! i~ a

G-invarillot mll.p. Therefl!lre the eigenvalues ef the actien .f fi en

( 11 )

N(Ag ) are the slime as these .f its action on N(Xg ), and t he cerrespending

eig.nbundle~ N~ abe corre1lpend under i'~ "5 in (11). 'Ire therefsre

deduce frem the definition sf L' (§2 (27») that

L' (g,A) i'* L'( g,X).

If we put thi~ i nte (10) and calculate as in t he n.n-equivariant

ca~e (§2), we find:

< i ''''L' (g,X), [Ali] >

< L' (Ii, X) , i; [A g J >

<L'( g ,X), Dxs[(flxg)*o) >

(here 85 in §2 l'Ie UM that the hemolcgy clallll [rl(p)] for a map ef

t he fGrm sf §2 (10) is the Poincare dual sf the cla1l II f* (J, " here (J

i s the generator sf HieS i ) )

(12 )

- 17 -

(fram the definitian .f F.incare duality)

<L' (g, X) U j*f*a, [X6J>

<j*f*aU l' (g,X), [Xg J>

(ane oan interchang$ sinoe 1'(g,X) and Xg ar$ even- .r add-dimen~i~nal Classe5 Bccarding as X is, and therefare have the srune parity, sa the

a:xpre!l!lian is zer. unle!l!l i = deg a '" dsg j*f*a il'l even)

<j*f*o,L'(g,x)n [Xg ]>

<j*f*a, Dxg (L I (g,X))>

<f*a,j.Dxg(L '(g,X))>

<f*a, DX(j!1'(g,X»)>

<f*o, Leg,X) n [X]>

<L(g,X) Uf*a, [X]> ( 13)

where in the lal'lt liMe we hAve !\ub1Jtituted the defini tiflns af j! and

af L(g,X) given in §2 and §1, re5peotively. In thi:; calculatian we

have Rat specified the a.efficient:;, but if an~ fall.,,:; threugh the

:!It.pe with the definitian:; fram §2, part, (I) in mind, ane find!l tha.t

the calculatian i:!l can:!li!ltent al~. when twisted caefficient:; must be

u:!Ied (Le. when Xg ilnd hence At il'l nan-arientable). In the last Une

all the cahamal.gy and hamalagy clal'lse5 appearing h~ve !limple c.efficients.

We new l'Iubstitute (13) inh (10), .btaining

1 TGT

If we further l'Iub~titute [X/G)

1 < ( -- rr*L) Uf+<o [X]> deg 7r •

E <L(g,X)U f *a, [X]>. gE ~

< (TiT L ,L(g,X) ) V f*a, [XJ >, g€ ~

and the de!lired equality (eq. (2) ~f §1) fallaw~ fr.m thi" equatien

and the fact thilt (9) d.efine~ the 1-cla~!S L = L(X/G) uni(]uely.

( 14)

(15)

It is interesting to look more closely at this equation and see

the relation between the properties of the equivariant classes L(g,X)

and the ordinary 1-olass. These properties of L(g,X) will be used

- 18 -

later. They also serve to make Theorem 1 of' §1 more plauaible by

ahowing that §1(2) defines uniquely a class L{X/G) € H*(X/G) which

has the properties expected of an L-class (leading coefficient 1,

zero in dimensions *4k, etc.). The properties in question are:

i) If g,h ( G, then

h"1(g,X)

ii) If G acts ef'fecti vely and G is connected, then the component

of L(g,X) in HO(X) is 0 if g*1, 1 if g=1.

iii) For all g (G, the component of L(g,X) + L(g-"-,X) in Hi(X)

ie zero unless ~ is divisible by four.

iV) L(g,X)[X] Sign(g,X).

v) The sum EgEG L(g,X) is in H*(X;~) C H*(X;¢).

( 16)

( 17)

It follows from i) that the average over G of the classes L(g,X)

is invariant under the action of G on H*(X), and therefore (by the

isomorphism (4) of §1) that it is ~* of a uni0ue element of H*(X/G).

If' we write this element as Ljdeg rr, then it follows from ii) that L

has leading coefficient 1, from iii) that L is non-zero only in

dimensions 4k, from iv) that L[X] Sign(X), and from v) that L is

a rational cohOmology class. Thus L has all the properties reouired

if it is to be equal to L(X/~).

The proofs of i)-iv) are quite simple. Property iv) follows from

the G-signature theorem (§2(28») and the definitiDn of j!. Property ii)

is clear sinCe the map j! railles dimensions by the difference of the

dimensions of its domain and target manifolds, so j,L'(g,X) Can have a

zero-dimensional component only if d.im xe, : X, whi~h for X connected

can only ocour if Xe:X and therefore (since the action is effective)

if g:1. To prove i), we observe that the map h:X" X defined by the

action of h lOG maps Xg isomorphically onto Xe ' (g'=hgh-l.) and that , the map h* pulls back the nDrmal bundle Ne to Ng (as G-bundles, i.e.

the splitting into eigenbundles is also pulled back). It follows from

this and from the functoriality of' the characteristic classes L, L~ and

La appearing in the definition of L'(g,X) that

L'(e,X) (18 )

Moreover, since h is an isomorphism we have h* (h- 1 )" and therefore

- 19 -

equation (16) follows from equation (18). To prove iii), we note that

the elements S and g-~ have the same fixed-point sets , and that the

eigenvalue decompositions of the commOn normal bundle Ng are related by -1 -1

N~ ,. Ng". In particular Ng ~ Ng We substitute this into the tI -" 1T 1T

definition §2(27) of 1'(g,X), using the fact that j, increasew dimensions

by dim X - dim Xg and the fact that L eNg) = e(Ne)L(Ng)-l only has 1T 11 1T 1T

components in dimensions = dim Ng (mod 4). We find that the proof of iii) 1T

reduces to showing that

E

only has components in degrees equal (modulo 4) to

dim X - dim Xg - dimNg 11

But this follows easily from the identity coth(x.-i8) = -coth(-x.+i9). J J

'{fe willmt give a complete proof of property v) here, since in any case

it follows from Theorem 1 of §1 and the rationality of 1(X/~). The

method of proof is to write

l: L(g,X) g~~

where, for Y a connected closed submanifold of X, 1y denotes the sum over

~ of the contribution from Y to 1(g,X) (i.e. zero unless Y is a component

of Xg, and jt[1(Y) ~ 1~(N~)] if Y is such a component, where j is the

inclusion of Y in X and Ng is the aie eigenbundle of the action of ~ on o the normal bundle of Y in X). This makes sense, since 1y is zero for

all but finitely many 3ubmanifolds Y. One then can show that each of

the classes 1y E H* (X) is a rational cohomology class by a ~alois-theory

type of argument. The argument when Y is a single point !xl is given

in the introduction to Chapter III.

- 20 -

§*. ~ definition 2! ~ ~ rational homology manifolds

The reason that the L-class can be defined for a rational homolo~y

manifold X is that the L-clasE is related to the signature of certain

submanifolds of X, and that there are enough of these sub manifolds to

determine L(x:) completely. It is reasonable to ask whether the equivariant

t-class has similar properties which allow its definition for rational

homology G-manifolds.

We cannot expect such a definition for the Atiyah-Singer class

L'(g,X), since it lies in the cohomology of Xg and therefore only can

be defined in terms of the local action of ~ near its fixed-point set,

which presupposes that this action is differentiable, or at least that

it looks like a differentiable action in a neighbourhood of Xg (cf.

Wall [tic], Ch. 14). But the class t(g,X) ~ 11* (X), defined in the

differentiable case as j!L'(g,X), can be characterised in certain

circumstances by a formula proved in the last section in the course

of proving Theorem of §1, namely (e (1. ( 13) of §3)

Sign(g,A) < L(g,X)U f*CJ, [XJ > ,

where A = f-~(p), £ being a map from X to S1 whiCh is ~-equivariant (G acts trivially on Si), :2 a sufficiently general point of Si, and (J

a generator of Hi(Si;z:). Equivalently, if we use the fact that f*a

i 8 the Poincare dual in X of the homology cla 55 i .. [AJ (this was used

in the proof siven in §3), we can consider (1) as saying that the

value of L(g,X) on a given homology class is Sign(g,A), where A is represents this class and which

a G-invariant submanifold of X which is the invers6 image of a point . A

for some map X~ S~ (in the differentiable case, this says that A ha5

tri viill normal bundle in X).

We thus wish to define a class

L(g,X) E H*(X;C) (2 )

for a rational homology G-manifold X in such a way that (1) still holds.

For this we require that there are enough "C.-invariant submanifolds A

with trivial normal bundle" in the sense defined above, i.e. that such

manifolds exist in enough homology olasses of X to determine 1(g,X)

completely, and also that there are not too many, sO that the conditions

(1) do not oonflict with one another. We cannot expect that all of the

- 21 -

elements of H.(X) are represented by embedded manifolds A of the type

desired (8~ was the case for Milnor's definition, at least in rational

homology), since a G-invariant submanifold A of X can certainly only

represent a G-invariant homology class. Therefore (1) only tells us G how to evaluate on elements of H.(X) , or rather only on those elements

of H.(X)G which Can be represented by good 8ubmanifolds A. To have a

reasonable hold on the group H.OQG, we must assume that G is finite,

in which case this group is isomorphio to H. (X/G) under the map 1T*

(cf. (4) of §1). Then knOWing the value of a cohomology class only

on the G-invariant part of H. (X) only determines the cohomology class

if it is itself G-invariant (for then it corresponds to an element

of H*(X/G) and is determined by its values on elements of H*(X/G»).

But we saw in §3 that

h*L (g,X) (all hE G)

in the differentiable oase, and it is easy to see that the same

formula will hold for a class L(g,X) defined using (1) (just

replaoe f in eq. (1) by foh, whioh is also a G-invariant map from

X to Si). Therefore if we want the oohomology olass L(g,X) to

be invariant under the action of G on H*(X), we should require that

h- 1 gh = g for all g,he:G, i.e. that G be abelian. Unlike the

requirement that G be finite, however, this does not limit the

generality of our definition, sinoe we want our olass L(g,X) to

share with the Atiyah-Singer class the property of depending only

on ~ and its action on X but not on G, and therefore we can always

replace G by the abelian subgroup generated by ~.

We can now state the theorem of this seotion:

Theorem 1: Let X be an oriented rational homology G-manifold, G finite abelian.

There is a unique class L(g,X) in H*(X;~)G satisfying (1) for all

simplicial G-inyariant lllB.pS f:X .... Si, and this olass agrees with the

class L(g,X) of §3 if X and the aotion of G are differentiable.

~: The last statement follows from the assertion about unique-

ness, since we saw in §3 that the differentiably defined olass L(g,X)

satisfies (1).

We now proceed as in the non-equivariant Case outlined in §2.

We work with complex coefficients, so that we are allowed to use the

isomorphism (4) of §1. We also assume that we have multiplied X with

a G-invariant sphere of large dimension to remove restriotions on the

- 22 -

dimensions of the cohomology groups to which we can apply Serre's

theorem. Then H*(x/G;~) is generated by cohomology classes which can

be represented by maps f:X/G ~ 5i • Such maps define G-invariant maps i -f:X ~ S by f=fow, and conversely any G-invariant map f factors through

X/G. The same applies to equivariant horootopies. He thus obtain a

commutative diagram in which all arrows are isomorphisms:

We now define, for each p .. Si, a map

Tp: "i(x)®~ ~~, Tp(f®A) = A5i gn(g,Ap )' (5)

where Ap f-~(p). We can prove that this is defined and independent

of E and of the choice of f. within its equivariant homotopy class. The

proof is just the same as in the non-equi variant case (Milnor [3t]). We

choose an open simplex ~i of 3i (Si has a fixed triangulation with

respeot to which f is simplicial) and, using Sard's theorem, a regular

value p E: Int(Lli ) of f.. Then there is a homeomorphism (given explicitly

in [31]) from f- 1 6i to ApX8i, commuting with the obvious maps to 6i , and

from the definition this is a G-homeomorphism if ! is G-equivariant. It

followa that T (f) = T ,(f) for almost all E and all E' close to.£. Then p p

a hOmotopy from .£ to a map £.' gives a cobordism from r1.(p) to f'-i(p) for

almost all £, and this is a G-cobordism if the homotopy is G-equivariant.

But the equivariant signature Sign(g,A) is an equivariant cobordism

invariant of A (cf. Ossa [33]), so we deduce that T (f) = T (f') for f,f' p P

equivariantly homotopic. Since f is equivariantly homotopic to ita

composition with any simplicial :utomorphism of gi, we can carry ~i onto

any desired i-simplex of without changing Tp(f). Therefore Tp(f) = T ,(f') for f,f' equivariantly homotopic and almost all p,p' (now without

p requiring that E' be close to .£). This shows that the map Tp is well-

defined, and that it is independent of E for almost all E. We denote

this common map by T. It is not hard to show that T is a homomorphism.

Using the isomorphisms (4) and Poincare duality in X/G (Which is a

rational homology manifold) we deduce the existence of L€H*(X/G) with 1 G T(f) = <IT*LUf~O',[X)>. Set L(g,X) .. -d--rr*r, E H*(X;~) •

eg 1T

- 23 -

§5. ~ formula !££ L(h'.X/G)

Now that we have defined L(g,X) for rational homology G-manifolds,

we find that the right-hand side of the formula for L(X/G) proved in §3

in the differentiable case is also defined when X is a rational homology

manifold, and we can ask whether it still gives the value of L(X/G).

This is the elise, and we even hlive the following more general result, which "'.

stated in somewhat other terms as Theorem 2 of §1:

Theorem 1: Let X be an oriented rational homology U-manifold, where U

is a finite abelian group, and let G be Ii subgroup of U. Then the

equivariant L-cllisses (in the sense of §4) of the induced action of U/G­

on the rlitional homology manifold X/e, are given by the formula:

TiT E L(u,X), UE ~

where e E U/G is a coset of G- and 1T the projection X .... X/G.

Proof: Equality (1) is modelled ~ter and proved using the corresponding

equation for the equivariant signatures, namely

de~ 1T Sign(e,A/G) = m E Sign(Il,A), UE e

where e is a coset of G in U and A is Ii U-manifold (or rational

homology U-manifold; (2) is Ii purely homological statement). This

is proved just as was the special CQBe U=G (eq. (1) of §3) by

applying to the ~ositive and negative eigenspaces of the middle

cohomology group of A the corresponding theorem for the traoes

tr(elV1 and tr(uIV) of the action of U on Ii vector space V and ot G

the induced action of U/G- on V •

(2)

To prove (1), we must show that the right-hand side is G-invariant

(so that (1) defines an element of H*(X/G)) and that the clase L(e,X/G)

that it defines satisfies the b.sic equation (1) of §4. But we

pointed out in §4 that equation (3) of that section holds also

when X is a r.tional homology manifold; it follows that the right­

hand side of (1) is G-invari.nt. (Here we do not need that U is

abelian, which implies h·L(u,X) = L(u,X) for hEG; it is sufficient

if G is a normal subgroup of U, in which case the actions of G- on

the left and right in (1) change the summands on the right-hand

aide but leave the whole sum invariant. Theorem 1 is thus also

- 24- -

true in this more general situation). Now we have to check that the

class L((,X/G) defined by (1) satisfies (1) of §4. Choose a (U/G)­invariant map f from X/C to Si and let A denote the inverse image of

a suffioiently general poi nt of

U-invariant map x~ Si and A the

that A "AlC. If we substitute

of L (!;, x/e) we can rewrite this

Si. Let f=f off denote the corresponding

inverse image _1_ ff [X]

i of ~ point under S , so

deg 1f • for (x/e] in the definition

definition as

1 /-deg 1r < 1T*L(~,X G) U 1T*f*a, [xJ > '" Sign{e,A).

and if we then substitute for IT·Lee,X/G) its value as given by (1). we

ob t ain as the equliltion to be proved

1 TGT l.: < L(u,X)Uf*a,rX ] > U€~

Signee,AlC).

But the elements s~TImed on the left-hand side of this are precisely

the numbers Sign(u,A), again by eq. (1) of §4., and therefore the

theorem has been reduced to the equality (2) given above.

(4)

- 25 -

§6. Applica.tion ts!.i,formula .2! Bott ~ ~ remarks £n ili.d2.

To illustrate the behaviour of the equivariant 1-ola58 L(g,X),

we will calculate it in a simple case, namely for linear actions on

X = Pn~' Since this is a smooth action, we Can calculate L(g,X) by

the Atiyah-Singer formula.

We write points of X as (Zo:"':Zn)' where (zo' ""zn) € S2n+1

is an (n+1)-tuple of complex numbers. Then the (n+1)-dimensional n+1 torus group T acts on X by coordinatewise multiplication, i.e. for

g = Tn+1 = S 1 x ••• x S 1

we def'ine the action on X by

We must calculate the fixed-point set of~. Clearly this is

Since at least one of the numbers 2i is non-zero, the complex

number C ill uniquely determined by Z E X and must belong to the

finite subset rCO' ""snl of 31• Therefore we can write Xg as a

finite disjoint union

where

and

Theref'ore X(C) is isomorphic to a projective space F (~), where 8+1 S

is the number of indices i with 'i=~ (we set s=-1 in the case (6».

In particuhr X(C) is connected whenever it i5 nonempty, so that (4)

is precisely the decomposition of Xg as the finite disjoint union of

its connectedness components.

(2)

(6)

- 26 -

The value of L(g,X) i5 now given by the Atiy~h-Singer recipe ~s a

sum over the components; we now calculate the contribution L(g,X)~ to

L(g,X) from a given component X(O. We let!!. denote the (complex)

dimension of X(,), and for convenience renumber the coordinates so that

then

;:,: Pit, s

and we shall use this isomorphism to identify X«) and F ¢ without 5

further comment. Let

X IS ll(X), y

(8)

be the usual generators of the oohomology of complex projeotive space.

Thus Y = c1(H) where H is the Hopf bundle over xC,). Since the normal

bundle of X(,) in X consists of n-s copies of the Hopf bundle, its

Chern class is (1+y)n-s . Moreover, if we identify this normal bundle Ng

with a tubular neighbourhood of X(,) in Xt we see that the action of ~

is given by multiplication with ,-1~i on the ith copy of H. Indeed

the fibre Ng of Ng at ZEX is identified with t n- s by the oorrespondence z

(Zo:···:z :Y1 : ... :Y ) 8 n-8

and the action of ~ on N~ is therefore

..... go(zo:"':z :Y1:"':Y ) B n-8

= «(OzO; ••• :, z : ( 1Y1: ••• ~ ~ y ) 8 8 5+ n n-s

(zo:"':z :(-'( 1Y1:···:(-1( Y. ) s· s+ n n-s ( -1 -1,.) ( (5+1 Y1' •• 0" ~n Yn- s •

As w~s pointed out in §2, the Atiyah-Singer characteristic class L~(~)

can be obtained from the same formula as that giving Le(~) if e is Il

complex bundle splitting up as a sum of complex line bundles. In our

case Ng splits up into Il sum of n-s complex line bundles in an

equivarillnt way, and each line bundle has characteristic class ~,

- 27 -

while X({) itselr has total Chern class (1+y)5+1 and therefore

L-01as5 (y/tanh y)5+1. Thererore the Atiyah-Singer formula (eq, (27)

of §2) givell for the class L'(g,X)C E H*(X(s) the value

L' (g, X) {

(y/tanh y)S+1 n n

j=s+1 ,.-1~ .e2y - 1 " "J

Now it is clear that the Poincare dual of yr!; J.fr(x({»

the homology clalls represented by the submanifold Ps-r~'

( 10)

is preCisely

If .1 denotes

the inclusion X(O C X, we have j~[P ~J" [P It], where the ~ s-r s-r

pight-hand lIide denotes the homology class in X represented by the

submanifold pte p t. The Poincare dual of this homology class is . s-r n n-5+r 2n-2s+2r C) then lon turn equal to x E 11 X. This shows that the

Gysin homomorphism j! is gi ve n by

n-s+r x (11 )

Thus to obtain L(g,X), from L1(g,X)\: we must replace :I. by 2S. in

formula (10) and then multiply the whole expression by xn- s This gives

5+1 Cii: .e2X +

Leg,X), ( ta~ x ) IT (X ct/e2x _ ) j=:H1 J

n ( IT x j=O

+ 1 )

- 1

where in the last line we have used the equdity 'j=' for j=O, .•• ,s.

Equation (12) is symmetric in the various coordinates, SO the fact

( 12)

that we renumbered the coordinates at the beginning of the calculation

does not matter, and (12) gives the desired contribution to Leg,X) from

the component X(C). If we now sum over the components, i.e. over all

C E 81 (that this is legitimate follows from the fact thOit (12) vanishes

if C • !,O' ···'(n]' since then each of the n+1 factors is a power series

in X beginning with a multiple of x rather than a constant term, and

xn+1 = 0 in H* ex)), we obtliin:

Theorem 1: Let g E Tn+1 act on X ~ P t by the action defined in (2), n

where {O""'{n E S1 are complex numbers of norm 1. Then the equivariant

- 28 -

L-cla85 L (g, X) is given in terIM of the Hopf class x ~ i(x) by

C" .6 2x n

( +

) Leg,X) l: n x ~ 2x , ~ 3 ' j=O C" .e -J

The :\um is in fact a finite one since the product appearing vanishes

in H*(X) if {~ l'O"""nJ•

Corollary: Let '" C 51 a denote the oyclic subgroup of a th roots

of unity, where ~ is a positive integer, and let

a i " = 1, i=O, ••.• nl -~

( 13)

( 14)

be a finite subgroup of Tn+1, acting on X = P nf, as in the theorem,

where aO, •••• an are pOsitive integers. Let £ denote the greatest

common divisor of the integers ~i' Let p: X ~ X/G be the projection

map and all other notations as in Theore~ 1. Then

p*L(X/G) = 1 d

n a .x ~ J n 7"ta-nh'7""lC ..... a ..,. (I-x-+ i':"":e:"'l):"'I')

O(~<u j=O J (15 )

Here the lIum over all reu numbers ~ between 0 and.". is in fact finite,

since the product vanishes unless aj~ m 0 (mod .".) for at least one ~.

~: Formula (15) was originally proved by Bott (not yet published).

It is quoted in Hirzebruch [14J (equation (1)) and a proof is given in [11].

Proof of corollary: By Theorem 1 of §1 we have

p*L(X/G) = drgrP r. L(g,X), gEG

~ and the factor iGI is simply the reciprocal of the order of the

subgroup H of G of elements acting trivially on X. Clearly

( 16)

and it follows from the definition of G th~t H = '"d' where £ is the

greatest common divisor of "'0' ••.• an• We can evaluate the sum in (16)

by using the trigonometric identity

.. z + 1 a---za - 1

( 17)

(which is proved by observing that the two sides have the same poles

- 29 -

and the same residues). Thus

E L(g,X) g"G-

I ~ E S1

~ ( X j=O

~ (a..X j=O J

.) C 1 , 2x t +

'--' C1(; .r/x a. J

(; . J=1 J

-a. 2a·x ( J e J -a· 2,,·x

C J e J

- 1 )

( Hl)

This sum is transformed into the one occurring in (15) by the substi­-2ie tution ,= e ,and the corollary is therefore proved.

We could of course write down an analogous formula for the

equivariant L-class of the action of an element of Tn+1/G- on the

quotient space pnt/G- by using Theorem 2 instead of Theorem 1 of §1.

We will give a second application of Theorem 1 in §15. But the

theorem is interesting for its own sake as well as for its applications,

since it illustrates the behaviour of the equivariant L-ola5s.

The first fact illul!trated by (13) is that, even in the cllse of a

differentiable action, L(g,X) E H*(X;¢) need not be II continuous function

of £. In II way this is surprising, since L(g,X) is defined using the

equivariant signatures Sign(g,A), and the equivariant signature not only

varies continuously but actulilly remllins constant as one varies ~ (since

it i~ determined by the action of g* in coho~ology). However, the sub­

mllnifolds A on which L(g,X) must be evaluated them8elve~ depend on ~

(it is insufficient to know the value of L(g,X) only on G-invariant

5ubmanifolds if G is .m infinite group), am therefore one cannot

conclude that L(g,X) varies continuously. Indeed, we see that (13)

has discontinuities at points gEG where two of the (i's are equal

for two reasons. First of all, the set [(0' ••• '{n l over which the sum

in (13) is to be taken becomes smaller when two of the 'i's coalesce;

this discontinuity would not be there if one counted each L(g,X), with ~ multiplicity equal to the number of 'i's with <i=~. But even

- 30 -

if one were to do this, the right-hand side of (13) would still be

discontinuous as a function of (SO' ""sn) € Tn'i, becau~e the function

x ~-i>ie2X + 1

Cl.Cje 2X -1 (19 )

is not continuous as a function of S at S = (j' Indeed, although for

each fi~ed ( this is a continuous function (power series) in!, its

constant term (obtained by setting x=O) is zero if (= (j and one if

( = Sj'

It is therefore certainly not the case that, for X a rational

homology manifold, the function L(g,X) defined in §4 for g € G'

!elements of finite order in Gj is a continuous function from G' to

H*(X;~). Therefore our first idea of extending the definition of

L(g,X) to all of G by using the denseness of G' cannot work. Never­

theless, it may be possible to define L(g,X) by continuity. We state a

Con,jecture! Let G be a compact abelian Lie grouP acting on a rational

homology manifold X, G' the dense subgroup of elements of finite order,

U the open subset ! geG I xg=xCI, and U' = un G'. Then the function

U' ~ H*(X;~) defined by the class L(g,X) of §4 is continuous.

The point about this conjecture is that, one the one hand, it holds

for differentiable actioM and therefore has some hope of being true,

and, on the other hand, it is strong enough to permit a consistent

definition of 1(g,X) on all of G if it holds. To see the first state­

ment, observe that the definition of I;(g,X) of §4 agrees with the

previous definition §1(3) if X is a smooth G-rnanifold, and the Atiyah­

Singer class L' (g, X) clearly is a continuous function of ;:. as long as

the fixed-point set does not change. To see how the conjecture's

truth would lead to a definition of L(g,X) for all gEG, we make the

additional assumption that the action of G on X has only finitely many

isotropy groups Gx (if X is an integral homology manifold, for instance,

this is known to hold: cf. Borel [2J, Ch. VI), Let G1, ••• ,Gk be the

various isotropy groups occurring, and f'or each subset Ie 11, .•. ,kl define G(l) = n iE! Gi , U{I) = !geGI gEGi ~ iEIj, and XCI)

!xEXI Gx=Gi for some ieIj. Then G is the disjoint union of the 2k

sets UCI), so it suffices to define L(g,X) for geU(I). But G(I) is a

olosed subgroup of G, and u(r) = [gcG(I) 1 xg XCI) I is for C(I)

precisely the set U of the conjecture, so if the conjecture holds, the

L(g,X) of §4 is defined and continuous on a dense subset UCI)' of UCI)

and can be extended to all of UtI). As before, we note that assuming

- 31 -

that G be abelian is harmless, since for arbitrary compact G we can

define L (g, X) by considering only the closed subgroup of G generated

by ~.

As well as revealing the discontinuity of the equivariant L-clase

L(g,X), equation (13) is interesting because of a certain formal

similarity with the main result of §5. To make this more epperant, . , . 2n+1 we rename the spaces and groups lnvolved. ,ve now wrlte X for S ,

. n+1 n+1 1 1 considered as the unit sphere ln C ,let U = T = S x .•• xS with

the obvious action on X, and let G be the circle group S 1 embedded in

U as the diagonal ((, ••• ,s). Then the quotient X/G = PnC is the space (1),

and equation (13) can be restated in the form

Lee,X/C)

where

f(u)

E f(u) UE"e

n 2x IT (x Sje + 1)

j=O ('je 2X - 1

Equation (20) is very similar to §5(2), which states that

rr*L(e,X/G) '" 1: L(u,X) UE"e

(20)

(21)

(22)

if G is finite and acts effectively on a rational homology manifold X.

We might therefore oonjecture that a formula like (20) hold.s in general,

at least for G a compact abelian Lie group, if G acts freely (so that

X/G is a manifold) or if only a finite nllmber of elements of G have

fixed. pOints (e.g. for a semi-free S1-action). The class f(u) would

then presumably satisfy "*f(u) = L(u,X), making (22) a oonsequence

of (20), and would also have to be such that feu) = 0 for all but

finitely many ~ in each coset e E U/G (so that the sum (22) makes

sense). I do not know if this is in general true. It is if G is

finite abelian, since then L(u,X) is invariant under G and we can

simply take feu) = 1T*-lL(u,X). It also is possible to define feu)

if the action is differentiable, since then (X/G)e is the disjoint

union of the sets XU/G with U€~ (and all but finitely many of these

sets are empty) just as in (4), and so we can take for f(u) the

oontribution in the Atiyah-Singer formula for L(e,X/G) coming from

the component xU/e of the fixed-point set (X/G)e.

CHAPTER II: 1-C1ASSE;S OF SYMMETRIC FRODUCTS

In Chapter I we obtained a formula for the calculation of the L-class

of the quotii'lnt of a d.i.fferantiable manlfold by a smooth, orientation­

preserving action of a finite group. In this chapter, we apply thie to

the quotient Xn/s of the nth Cartesian product of X with itself by the n

symmetric group on n letters, i.e. to the nth symmetric product X(n) of X.

In order that the action of S on XU be orientation-preserving, it is n

necessary that X have even dimension 25.

We give in §7 a description of the rational cohomology of X(n), as

well as the various notation necessary for a statement of the formula

for L (X(n)), whioh is then bi ven in the following section. Since the

complete formula obtained is rather obscure, we also make several comments

explaining the meaning of the various terms ana of the formula as a whole.

The proof occupies the following four sections. Section 13 contains an

explicit evaluation of the formula in two ~pecial casell-- X a sphere of

even dimension, and X Ii. Riemann surface. In the latter case, X(n) is a

co:nplex manifold whose Chern class (and h0nce also 1-cla53) wa3 already

known (Macdonald [2~1); thus we are able to check our general formula.

In §14we consider the more general 5i tuation where a finite group G

acts On X. Then there is an induced action on X(n) (diagonal action).

and we can apply the result of §5 to calculate L(g,X(n)) for this action.

Since the oa.lculation is similar to that in the non-equiva.riant case, we

give the proof more briefly, in a single section. The formula obtained is

even more complicated than the non-equi variant one, but again we find tha t

the dependence of L (g,X(n)) on ~ is very siCiple--if the order of' '" is odd.

In the last section of the chapter we compute explicitly the form which

the equation for L(g,X(n)) takes if X is an even-dimensional sphere. If X

is 82 , then X(n) = P ~ and we can check the result with the computation n of L(g,Pn¢) made previously in connection with the theorem of Bott.

The most interesting fact whioh emerges is the very Ilimple dependence

of L(X(n)) on the number of factors II in the syw~etric pOwer. Namely, if

we oonsider the inclusion l of X(n) in X(n+1) induced by choosing a base­

point in X, and then form Q = j*1 (X(n+1) ).L (X(n»-~ (this is possible n

since L(X(n)) is invertible; Qn would be the L-class of the normal bundle

- 33 -

if X(n) were a smooth manifold smoothly embedded in X(n+1) ), it turns

out that Qn is independent of ~ in the sanse that j*Qn+1 = Qn' Thus

Qn is the restriction to X(n) of a class Q in the cohomology of X(oo),

Moreover, Q turns out to be in a certaineense independent of X. There

is namely a oanonically defined element ry E H2s(x(oo)) arising from

the orientation of X, and the class Q is equal to Qs(~) with Qsa power

series in one variable whose coefficients depend only on II = (dim X)/2.

For example, Q1(t) = t/tanh t; this corresponds to the case where X

is a Riemann surface, in which case X(n) is a smooth manifold and Qn really can be interpreted as the L-olass of the norlllBl bundl e of

X(n) C X(n+1). For ~ larger than 1, however, the power series Q (t) 5

does not split formally as a finite product rr~ 1 (x ./tanh x.), and we J; J J

deduce that the inolusion of X(n) in X(n+1) does not have a normal

bundle in the sense of Thorn.

In the e qui variant case, if £. acts on X and has finite order .E.

we get a similar result if .E is odd, namely the restriction to H*(X(n»

of r.(g,X(n+p)) equals the product of Leg,X(n) with a factor 0. which

is independent of B, of X, and of the Bction of £. on X, but depends

only on ~ and .E (it is again a power series in n, this time with

ooefficients depending on I'. as well as on~; for instance it equals

1)/tanh p~ if s=1). This result has two features of interest:

first, that there is a simple relation between L(X(n») and L(X(n+p» ,

so that there is a kind of periodicity with period E in t he eauivariant

structure of the symmetric products of X, and secondly, t hat there is a

different behaviour for elements of odd and even order (if E is even

there is no periodicity).

Sinoe the caloulations are long and perhaps unoonvinoing, I have

tried to give as many computations as possible f or which a known result

was available for comparison. Unfortunately, thi. S "'as only possible for

X two-dimensional, I'>ince the power series Qs(1)) for s>1 has not previously

occurred in this connection. As already mentione~ it was possible in two

cases: ~aodonald 's work on symmetric powers of Riemann surfaces for the

non-equivariant case, and the formula of §6 (heN X=S2) for the equivariant

case. In both cases, the previous result had been obtained by quite

different methods (Macdonald dOes not use the index theorem, and in §6

we did not use the fact that Pn~ is a symmetriC product) and in a Quite

different form requiring a long computation to be shown equal to our

result. Thus these verifications lend considerable credibi lity to the

theorems of this chapter.

- 34- -

§7. The rational cohomology 2! ~

If X is a compact, connected topological space and 2 a positive

integer, the nth symmetric product of X is the space

X(n) (1)

with the quotient topology, where Sn is the symmetric group on n letters

acting on the nth Cartesi~n product Xn of X with itself by permutation

of the factors. Thus Xen) is the set of all unordered n-tuples of points

of X, with the obvious topology. If n=O, we define Xen) to be a point.

The nth symmetric product of X is dso sometimes called its nth symmetric

power, and is variously denoted X(n), X[n], and SP~,

If we choose a base-point XOEX, then there is a natural inclusion

j: Xen} C X(n+1) (2)

sending an unordered n-tuple !x1, .•• ,xnlcx to its union !xo,x1, •• ·,xnl

with !xOl. However, there is no natural projection map from X(n+1) to

X(n), since there is no natural way to choose n elements from a set of

n+1 elements. We will also use the letter J to denote compositions

of tIE map (2) with itself, i.e. "ny inclusion X(n)C X(m) with m>n. Let

X (00) (3)

be the limit of the direct system defined by the maps j. and use j to

denote the inclusion of Xen) in X(=) also.

The purpose of this zection is to describe the (additive) rational

cohomology of X(n). This is very easy. uzing H*(X/Gi~) ~ H*(X;~)G and

elementary properties of cohomology (as found in Spanier), and this

seotion can be skipped except for the equations (8), (9), (17) giving

the notations used for the basis of' H* (X (n); ~) and for one easy

proposition (eq. (24»).

All cohomology in this seotion is to be understood with coefficients

in ~ (or ~ or ~; we only need to have a field of characteristic zero).

This results in two simplifications. First, as mentioned above, the

map "*:H*(X/G) ~ H*(X) is then an isomorphism onto H*(X)G (cf. §1);

this isomorphism will be used without explicit rrention to identify

H*(X/G) with H*(X)G C H*(X) for any G-5pace X with G a finite group.

- 35 -

Secondly, there is a natural isomorphism of H*(XxY) with H*(X) ® H*(Y).

Therefore

H"'(X) ~ ••• * H*(X)

has as a basis [f. x ••• xf. I i:l.'" -,in € IJ, where If; I i € IJ is an 1:1. J.n ~

additive basis for the finite-dimensional vector space H*(X) over ~.

Recall that the map in cohomology induced by the interchange mapping

T: XxY'" YxX

sends Vxu€ Hj+i(yxX) (where u€ If(x), v€ Hj(y) ) to (-1l j uxv€ Hi+jeXxY).

In particular, with X=Y we see that the map induced in cohomology by an

interchange of factors is not simply the corresponding interchange, but

contains a further factor -1 if two terms of odd degree are transposed.

It follows that, for a E S , the effect of cr. on H('(Xn) (where (J acts n

on Xn by a(x1, •• • ,Xn ) ... (xa (1)' ••• ,Xa(n))) is

(6)

where u1"", un in H* (X) are homogeneollS elements and v is the mlJIlber

of tran:sposi tions i+-+ j of the permutation a for which u l and u j have odd

degree (this number is well-defined, i.e. independent of the decomposition

of (J as a product of transposition5, modulo 2). Nrite 1T j :Xn -> X for the

projection onto the jth factor; then we can reformulate (6) as

where in the last line we have used graded commutativity. (One can

see this more easily by noting thl1t 'If,ioU = 1To(j)' so that U*(1TjU)

1T~(j)Uj') Therefore the symmetrization of u 1x, "xun is

r o"'(u1x., .xu ):::: ~ ";(1 )u1 _ •• ll~(n)un J<OS n a"S

n n

€ H*(Xn)~n = H*(X(n)).

Here and in the future, we om.i. t the symbol U and a.enote cup produ cts

simply by juxtaposition.

We choose a basis fa, ,."fb of H~(X) with fb=1 and each fi

(8)

- 36 -

homogeneous, say ot degree di , Then H*(X(n)) is spanned by the elements

<f. , .. "f. ~. Since this element is independent (up to sign) of the J~ In

order of the indices ji' we need only specify the number nj of times a

given index j (j=0,1 ••• "b) appears in the set Ij1 •••• 'jn l . Thus, let

<no(fO)···~(fb)~ <~Q' ••• ,fa:r1' .•.. f 1,' ···'rb' •• "fb? no ~1 .\b

€ H*(X(n)),

IIhere nO+" .~=n, There is, however, one more restriction if' we

wish to reve a basis: namely, ni must be ~1 if di~deg fi is odd.

Indeed, from (8) we see immediately that if. for example, u1"u2 and is

of odd degree. then <u1,.,.,un> is zero. Thus we have

Proposition 1: Let fO,. .. ,fb be a ho:nageneous basis f'illr H"'(X;Il)). Then

a bSl:lis for H* (Xen);~) is. given by the set of elements (9) with

nO+'''+1\ = nand ni ,,1 for all i with deg fi Odd.

CorOllary: The POincare polynomial

co

F(X(n)) = E xr dim~ ~(X(n)i~) r=O

of X(n) is given in terms of the Betti numbers

of X by the formula

/" t n peX(n)) n=O

[] (_1 -d)!%[] (1 + txd 'fd d~O 1 - tx d~O )

d even dodd

In particular (x= -1), the Euler characteristics of X(n) and X are

related by

Note: FGrmulas (12) and (13) are due to Maodonald ["2I1J.

Proof of corollary: By the proposition, we have

(10 )

(11 )

(12)

dim(\! Hr(X(n» = 'Ino, .... ~~OI nO+, .. +~:=n, ni ,,1 for di odd,

nodo + .. , + ~~ = rl,

from which we find:

- 37 -

OQ 00

E E tnxr dim ~(x(n» n=O r=O

no' ".,~"o n . .;1 for d. odd

:1. :1.

b (00 n nEt

j=O n=O

nd.) b ( 1 x J n E j=O n=O

d. even J

d j odd

b 1 ) b ( d) j~O C d' n 1 + tx J tx J j",O

d. even d. odd J J

n ndj ) t x

00 d -'d 00 d fld

n (1 - tx ) n (1+tx ) ::

d.O d=O d even dodd

,Ve now want to compare the cohomology of' X(n) with that of' X(n+1),

. t1.~· l' (2) Th . l' . I f Xn. xn+1 Xn uSlng u::: lnC USlon . e lnc US10n J 0 1n as x Xo inducell the map

if un+1

if' un+ 1 * 1

in cohom01ogy; therefore the map j* from H*(X(n+1) :: H*(Xn+1)Sn+1

to H*(X(n» = H$(Xn)Sn , which is simply the restriction of j'* to

the S i-invariant subspace of H*(Xn+1), operates by n+

'\' ---------

(15 )

j* <u1 ' ••• , U 1> =) rr" (1 ) (u1 ) .•• rr* 1 (u - ~ ( 1 ) ) ••• rr" ( 1 ) (u 1 ) n+ '....-1 a n+ (J n+ a n+ n+ o E 50 +1

Ua-l (n+1 t 1

011 » I...--.i ~

j= 1 O€Sn+1 u j =1 0'(j)=n+1

n+1 r;

j~1

u j = 1

- 38 -

where aa usual the A indicates the omission of a factor. Therefore

[ nb<no(fo)"o'~_1(fb_1)~-1(fb» if ~>O, if nb=O,

or

Thus if we renormalize the element (9) by dividing by ~!, we obtain

an element which is stable under j". IVe therefore define

E H*(X(n),

where in the caae n~ nO+" '+~-1 we have defined ~ as n-nO-' "-~-1'

Then (16) states

In other words, the sequence [nO(fO)"'~_1(fb_1)]n (n;1.2 .... )

defines an element

lim l!*(X(n) n

such that

( 18)

( 19)

for the inclusion j: X(n) C X(oo). Because of relation (20), we will

omi t the subscript.!! in future, leaving the question whether the stable

element or its restriction to SOme X(n) is meant to be decided by the

context. Noti.ce that, sincB f O'" .,f'b_1 have peait1ve a.srees, there

are only finitely many elements (19) of given degree, so that each

of the groups Hj (X (00)) is f:i,ni to-dimensional over «l (though infinitely

many of them are non-zero). In particular X(oo) has a well..(W.fined

POinoare power series, which from eq. (12) above is clearly

p(X(co)) ; IT IT ( 1 + x d)iJd (21) d>O, even d>O, odd

- 39 -

We have 100ked especially closely at the role of fb= 1 in expression

(9) above. However, there are other f. IS in (9) which act in a special 1

way, namely tho5e of the highest degree occurring (or, more generelly,

any f. such that f.fj=O for all f. of positive degree). Assume that fO 1 L J

is such an element, and that its degree ~ i s ~ (this is the only

CB5e we will need later). Define

(22)

Then

In the expression on the right, we interchange the summations and replace

i by j=a(i), obtaining (since f a iB of even degree and. commutes with every­

thing) n

E E (~~(1)U1)"'(~~(j)(ujfo» •.• (n~(n)un)' O€Sn j ,, 1

But ujfo is zere unless u j =1, by our assumption on fa, so this is

j~1 L a;s (1T~(1)U1)"'(~~(j/O)"'(~~(n)un) ] u.= 1 n

J n

Z <u 1, .. ·,u. 1'£:0'u. 1'''·'un>. j= 1 J- J+

u . = 1 J

Therefore (we use an overline rather th,m brackets to avoid ounbigui ty)

or, finally,

Clearly the sequence ~ (n=1,2, ..• ) define~ an element ~ € HdeX(=» and n

(23) is a stable relation. We have thus proved

Proposition 2: Let deg fa = a. be even ana. a;.di for all i. Then nO

[nO(fo ) n1(f1)"'~_ 1 (fb_1)] 1] [n1(f1)".~_ 1 (fb_1)] (24)

in H*(X(oo)), where 1] = [1(fo)] E HdeX(oo).

- 40 -

§S. statemsnt ~ di5cussion of th8 formula for ~)

We now as~ume that the space X whose symmetric power5 are being

studied is a closed, connected, oriented differentiHble manifold. Then

the action of the symmetric group Sn on xn is smooth; the condition

that xn/S n be a rational homology ma.nifold is that this &ction is also

orientation-preserving. Sinc8 Sn is generated by transpositions (for

n > 1), this i5 the case eu.ctly when the interchange mllp T:XxX .... XxX

is orientation-preserving. Because of the graded commutativity, this

holds if and only if the dimension of X i5 an even number 25.

We let

be the Poincare dual of [pt. ] € Ho(X). We then choose a basis for

H* (X) as in §7, i .. e. fO •••• ,fb with each fi homogeneou~ and with

fO= Z and fb = 1. We also introduce II second basis

which is dual to !fil under the intersection form, that is,

<a.f . • [X]> l J

o .. lJ

( i,j" 0,1 , ... ,b),

Thus 8i is homogeneous of degree 2s-di , where di " deg f i ,

The result of the last section was that a basi5 for H*(X(oo») is

given by the elementoS

where the square brackets have the meaning given in §7 and where

1] =

is the element whose restriction to X{n) is

(1 )

(2)

(3)

(6)

The map j*:H*(X(oo) -+ H*(X(n» is surjective, and a hsis for H*(X(n))

is given by the images under j* of the elements (4) for which

no + ••• + ~-1 ~ n.

- 41 -

W. now ~tate the main theorem of thi~ chapter.

Theorem 1: Let X be a connected, closed, oriented differentiable

manifold of dimension 2s. Let ry be the element defined in (5) and (6). Let l denote the inclusion of X(n) in X(n+1). Then

j*L(X(n+1» = Q (11 ). L(X(n), s n

where Qs(-) is the power series in one variable defined below, and

depends on ! but not on the space X or the number of factors~. An

eqUivalent statement is that there exi~t~ an element G of H**(X(oo»

such that (if we use.J. also to denote the inclusion X(n)CX(oo»

L(X(n»

(7)

(8 )

The power series Qs is defined as the unique even power series such that

coefficient of t 2k in Q (t)2k+1 s

It can also be defined as

t

r:m-where

g (t) ::: t s

(k~O) •

( 10)

(11 )

The first few terms of these power series are given in the table at the

end of this section.

The elas/! GeH**(X(oo» is defined as follows: Let B be the graded.

pofllr sen-s ring over GJ in variables to"'" \-1 with ti of degree di'

i.e. B is the quotient of GJ[[to' ""\-1]] by the relations

d.d. titj ::: (-1) ~ J tjti (i,j ::: O,1, •.• ,b-1). (12)

Let A be the graded tensor product of H**(X) with B. We use ~ to

denote e81 (for eE H**(X») and ti to denote 1®ti (i=O,. •• ,b-1); then

dn. (-1) -:L t.e

].

in A. We define a e A by

- 42 -

Since B is a paNer series ring rather than a polynomial ring and we

tensored with H**(X) instead of H*(X) in forming A, it is legitimate

to form power series (with rational coefficients) of elements in Ai

thus gr(iX) e A is well-defined. We write Lc(I) for the component

of degree 2c of the L-class of X (~ deviates from ~ ~ notation

where Lc denotes a certain polynomial in Pontrjagin classes of total

degree 4c), or for the corresponding element Lc(X)~1 of A. There is

a linear map < -, [X]> from the graded ring A to the graded ring B,

defined on the generators e t .... ti of A by 11 k

< e t .... t. ,[X» 11 J.k

<e,[X]>

Set

H 1;·cS __ O <15 1 (a)L (X), [X]> 8+ -c c e B ( 16)

and define numbers as the coefficients in the expansion

H

e B,

where e(X) is the Euler characteristic, etl denotes exponentiation, and

the sum is over all integers nO' ... '~_1~0. Then

G

the summation being as in (17), where Q (-) and f (-) are the power s 5

series defined above and f~(-) is the (formal) derivative of f s (-).

Before discussing the interpretation or significance of Theorem 1,

we shoul d perhaps make some remarks elucidating the formalism, since

the complete statement is somewh2t involved and perhaps confusing.

In the first place, the essenti21 and most interesting assertion

of the theorem is the relatively simple statement (7), either as a pure

existence theorem (namely the existence of a power series Qs independent

of ~ for which (7) holds) or, combined with the formula for Qs given in

eq. (9) Or in eqs. (10)-(11), as a preCise statement of the relstionship

between the values of L(X(n» for two successive values of~. The

interest of this relation lies in a certain formal resemblance to a

- 43 -

similar relation involving the L-01&ss of a normal bundle; this will

be discussed below. As sts.ted in the theorem, equation (7) is

equivalent to the existence of a cohomology class G in X(oo) such that (8)

holds for every~. To see this, we notice that (8) trivially implies (7)

since the inclusion of X{n) in x (co ) is the composition X(n) C X(n+1) C

X(=). Conversely, if (7) holds for every~, we define a class Gn in

H*(X(n» as Qs(~-n-1 LlX(n» and deduce from (7) that j*Gn+1 = Gn ,

which means that the sequence !Gnl:=1 defines an element G of the

inverse limit lim H* (X(n» = H* (x("",» with j*G = G for all n. Thus , n -we can break the theorem up into the main statement, that L(X(n»

behaves exponentially as a function of ~ (i.e. is of the form Qn+1 G with Q, G independent of ~), the statement that Q is given by the

expressions (9)-(11), and the statement that G is given by the proce­

dure stated in the remainder of the theorem.

The definition of the power series Q (t) = 1 + t 2/3s + s

(the first few coef'ficients of Qs and various related power series are

given in the table at the end of this section) requires little comment.

The equivalence of the definition (9) and the definition given in eqs.

(10) and (11) is of course an exact parallel to the characterisation

of the function

(19 )

given by Hirzebruch ([11], Lemma1.S:1), and the proof is exaotly the same.

The functions 0oCt) and Q1(t) can be written in closed form (QoCt) is

equal to [1 + J1+4tS]/2 ) but the series Qs(t) for s~2 is not an

elementary function. Note that, since Q (t) is an even power series s

and TJ has degree 2s, equation (7) implies that L(X(n») and LCX(n+1)

agree up to degree 4s (or more precisely, j*L.(X(n+1) = L. (Xen») for :1 :1

i < 2s). so the value of L(X(n» is essentially independent of ~ in

degrees smaller than 2,dim X.

Even less need be said to explain the definition of G, though

this is more complicated. We only remark that, since gr(t) is a

power series with no oonstant term, H oonsidered as a power series in

the ti's also has no constant term, so it is legitimate to form the

power series eH in equation (17).

It would be very pleasing to have a simple proof of equation (7),

or equivalently of (8) without an explicit evaluation of the power

series G. One way

L(X(n+1») and show

- 44- -

to do this might be to assume given the class

directly that Q ('7n) -~ j"'L (X(n+1») satisfies s

Milnor's definition for the L-class of xen).

However, despite the rather cumbersome formula for G, it is

pOl!sible in certain situations to use the entire formula (8) to

obtai» completely explioit formulas for L(X(n». This is the case

when X has especially simple cohomology and L-olaS5, for example if

it is a sphere (of even dimension) or a Riemann surface (for these

two cllses the caloullltion will be given in §1;). Moreover, it is

possible to write G in a shorter and more natural way--in particular,

in such a way that it does not involve the choice of a basis for H*(X)-­

and indeed it is this basis-free version whioh will emerge from the

proof in §§9-12. We preferred the more unwieldy form given here

because the basis-free formulation requires the introduotion of yet

more "bstract notation which would have _de the statement of the

theorem even more obscure, and because one is in any ca~e obliged to

expand in terms of a basis for concrete applications such 8S those H

mentioned above (indeed, even the power series gr(a) and e used to

shorten the statement of the theorem given here must be expanded in

the course of an explicit calculation).

We can use the evaluation of G to obtain complete results for

L(X(n» not only when X is a simple manifold, but also for arbitrary X

and small values of n. This is slightly easier using the basis-free

form of ~heorem 1 but can also be done with the form given here. For

n=1 we Can easily check that Theorem really dOes give L(X) as the

value of L(X(1)). For n=2 we obtain (with 7TlXXX -+ X(2) the projection)

7T*L(X(2) ) L(X)xL(X) + e(X):<Ix z. (20)

In particular, the signature of the symmetric square of X equals

[(Sign X)2 + e(X)]/2, a special case of a formula of HirBebruch for

Sign(X(n) which will be proved in §9 (and deduced from the formula

for L(X(n)) as a check on the latter in §13). For n~3 the formula

for IT*L(X(n)) is rather more complicated; it is a certain polynomiill

in the lifts to H*(Xn) of the elements Z E riseX), L(X) E H"'(X), and

a e; H2s (XxX), this last being the restriction to XxX of the Thorn class U

i)1 H2s (XxX,XxX-L:.) (where t:, is the diagonal in XxX).

The fact that L(X(n)) is given--when X is a differentiable

- 45 -

manifold--by a certain polynomial in the classes ~, Lc(X), and!

and their lifts to H*CXn) makes it clear that Theorem 1 still makes

sense when X ie a rational homology manifold. Indeed, the cohomology

classes ~ and ~ are defined for any space satisfying Poincare duality

(i.e. with the global homological properties of an oriented differen­

tiable manifold), while L(X) is defined for a rational homology

manifold by the definition of Thorn or of Milnor. In the spirit of

Chapter I, we ask whether a result involving L-classell which is known

to hold for differentiable manifolds and which inVOlves no differentiable

data (normal bundles, eigenvalues, etc.) also holds for rational

homology manifOlds. If we look at the proof of Theorem 1, we find that

only at one place did we use the differentiability, namely an applica­

tion of the G-signature theorem of Atiyah and Singer which WRS used to

prove the following result:

Propo5ition 1: Let X be a compact, connected, closed, orientable

differentiable manifold of dimension 28. L~t a:Xr~ Xr be the

permutation map sending (x1, ••• ,xr ) to (x2, ••• ,xr ,x1), Let f:~~ S2k

be a smoot h map with r(aoy) '" r(¥) for all yfi Xr , and choose

p E S2k as a regular value of f.. and of f ill (where Ll C Xr is the

diagonal); thus A = f-~(p) i s a submanifold of Xr of dimension

2rs-2k and AntJ. is a manifold of dimension 2r-2k. Then

L-k ~ (X), ifr is even, k=O,

Sign(a,A) :: 0, if!: is even, k )O,

Sign (An l», ifr is odd .

Thus we have a statement, formulated purely in homological term5 and

with notions that make sense for X a rational homology manifold (we

(21)

l~t f be simplicial and interpret transversal ity in the sense of Thorn ) ,

but only known for X a smooth manifold. A proof of Proposition 1 for

<lny r<lt:lon<ll homology manifold X would prove that Theorem 1 also holds

for X, since the other steps in §§9-12 cllrry over without ohange. There

is lin analogous proposition if a finite group acts on X whose truth

would imply the validity of the result of §14 for L( g, X(n») for X a

rational homology manifold.

We clo5e thi.s section with a discussion of equation (7) and its

above-mentioned resemblance to a formula involving normal bundles.

Recall (p. 1) that Thorn showed how to define a bundle v over a

- 46 -

rational homology manifold embedded in a certain way (one says that N is

a submanifold ~ orthogonal ~ structure) in another rational

homology manifold M. A rational homology manifold M always has enough

such sUbmanifolds to generate its rational homology, If M is a smooth

manifold and N a smooth 5ubmanifold, then v is the usual normal bundle,

so we can consistently refer to v in general as a normal bundle. It is

a real vector bundle whose dimension equals the codimension of N in M. ~inally, Thorn showed that the 1-classes of M and N (in his sense) are

related to the L-class of v just as in the differentiable Caee, i.e, by

j*1 (M) L(v)L(N), (22)

where j is the inclusion map from N to M.

Equations (7) and (22) are identical in form if we take for Nand

M the rational homology manifolds X(n) and X(n+1), and our first thought

is that this formal identity cannot be accidental but rather arises

because X(n) really is a rational homology submanifold of XCn+1) with

orthogonal normal structure, its normal bundle vn in the sense of Thorn

then inevitably satisfying

However, this is not the case if ~ is greater than one (if 5=1, X(n)

is a differentiable and even a complex manifold, and vn is the normal

bundle of X(n) in X(n+1) in the usual sense; the first Chern class of vn

equals ~ E H2 CX(n) and Q (~ ) = ~ /tanh ry is the usual L-ola55 n s n n n of a complex line bLlIJ.dle). To see this, we apply in reverse the

multiplicative sequence used to define the 1-class in terms of the

Pontrjagin olass (cf. Hirzebruch ['11,], §1, espeoially for the proof

that the multiplioative sequence associated to x/tanh x is invertible).

Since Q(ry) is a power series in ry2, it follows from the properties s

of multiplicative sequences that the sequence Rs (I) obtained is also

a power series in 1)2. But 1)2 has degree 45, and the bundle vn (if

it exists) has dimension 2s and therefore a Pontrjagin class cutting

off at dimension 4s. Therefor'e if (23) holds, we have from p(vn)

Rs(l)n) that the series Rs must break off after the second term:

But the multiplicative sequence x/tanh x appcied to (24) gives

as the corresponding L-class the series

- 47 -

where we have written ~k for the coefficient 22k (22k-1 -1)~(2k)! of 2k

x in x/sin 2x (this coefficient is denoted by sk in ~lJ), Since

Q~(I1) -to (...1. )112 -to (J...8 - 2s )T/4 -to •••• (26) ~ 3s 5 9

it oan only be of the form (25) if the relation holds which is obtained

by equating coefficients of T/ 2k in (25) and (26) and eliminating a, i.e.

~: - P2s 211'2

s 2.

It is easy to see th~t (27) is fulfilled only for s=1,

We have therefore proved that X(n) is not in general a rational

homology sub manifold of X(n+1) with orthogonal normal structure. However,

the formal analogy between (7) and (22) s t ill suggests that it has a

particularly nioe normal structure in sorne sense. We can hope to find

some more general type of normal bundle than O(n)-bundles, for whioh

(rational) Pontrjagin olasses or L-classes are still defined and such

that the inclusion of the nth symmetric product of a manifold in the ( ) st .. n+1 always has a normal bundle 1n the genera11sed sense. In fact,

a definition recently has been given for such a generalised bundle for

homology manifolds (Martin and Maunder [2~), These objects, called

"homology cobordism bundles," are (roughly) defined as projections E~B for

which the inverse image of a cell in B is H-cobordant to the product of

the cell with the fibre. the fibre being taken as Dn in an n-dimensional

bundle. It was shown (in the paper referred to above and in Maunder ~~)

that these objects form a reasonable category, allowing Whitney direct

sums, that they possess an 1-clas8, that the inclusion of N as a

homology submanifold in a homology manifold M always has a normal

homology cobordism bundle v and (22) is satisfied, and finally that

the set KH(X) of stable isomorphism classes of homology cobordism

bundles over X is an abelian group and that X~KH(X) is a representable

functor. These results were proved in the category of homology manifolds

but pos sibly still hold if' one only requires that the spaces be

rational homology manifolds, and replaces homOlogy by rational homology

in the definition of H-cobordism. Then equation (7)couldbe interpreted

as saying that the L-class of the rational homology cobordism bundle vn

of X(n) in X(n+1) is given by (23).

- 4tl -

In [l~, the Pontrjagin class of a homology cobordism bundle wa~

defined by using Thorn's definition of the rational Pontrjagin clalls of

a homology manifold (as the cla ss obtained by applying to the L-class

defined in §2 the inverse of the Hirzebruch roul tiplicati ve sequence

x/tanh x) and then requiring (22) or its analogue for Pontrjagin classes

to hold; this suffices to define L(v) for any bundle since a bundle CaR

be thought of as the normal bundle of its zero-S!JctiOA in its tot,"l

"pace. However, this seem15 to be the wrong generalisation of the

Pontrjagin clas~, since, as shown above, the Pontrjagin class defined

in this way for vn does not break off at the right pOint. Iadeed, it

almost certainly does not break off at all, since the formal power

series Q (ry) in a variable ry of degree 25 does not seem to split as '" s fini te product ni Q1(xi ) with variables Xi of degree 2. It seems more

appropriate to define the Pontrjagin class of our normal bundle vn by

p(lJ) = n (28)

If s=l, then vn is a complex line bundle over X(n), and (28) is the

usual Fontrjagin class. In general we call lJn a "1in~ bundle of type s"

(we omit in future th~ wora~ ura tional homology cobordism" before

"bundle"). The :standard model is obtained by taking for X a sphere of

dim~nsion 25. The fact that Qs (-) is independent of the number of

factors!! in the sy;nmetric product ~uggests that the bundle vn over

82s (n) ill itMlf independent of !!, i.e. that j*vn+1 '" 'vn ' In any case,

this certainly holds for the L-class as defined in (23) or the Pontrjagin

class as defined by (28 ). 'rYe then obtain a5 classifying space for line

bundles of type ~ the l imit l~m 328 (n) = 825 (00), over which there is a

universal line bundle of type ~ defined as the limit of the bundles vn'

and for this bundle v we have

L(v) p(v) (29)

where 1) is the gen~rator of H*(S2s(oo) ,~) = (l[[T/]]. We then define in

general a line bundle of type s to be the pull-back of v to a space X

by some map f:X ... 825 (00) , its 1-class and Pontrjagin class are then

defi ned as the pull-backs of the elements (29). Therefore there is an

isomorphism between the set of homotopy classes of maps! and the set

of isomorphism classes of bundles of type ~ over X. But a well-known

- 49 -

theorem of Dold and Thorn [9J states that the homotopy groups of an

infinite symmetric product X(=) are isomorphic to the homology group~

of X, and thus in particular that the space S2s(oo) is a K(~,2s). The

isomorphism cla88e~ of line bundles of type ~ are therefore in t:1

correspondence with the elements of' [X,K(lI,2s)] _ IiS (X;!:). Thus

a line bundle t over X ia completely classified by a "firllt Chern claslI" ( 2:s 25 C1 t) € H (X;~), where t:X ~ S (00) is the classifying map of e (i.e.

t ;: f'* l') •

All of these remarks are naturally conjectural only, depending on

the existence of an appropriate category of rational homology bundles

enjoying at least the properties of the Martin-Maunder bundles (namely

that direct 5um5 and induced bundles can be formed, that there is a

clasllifying space, and that the bundles have L-classes). We can go

further in the description of the form the theory might take if these

bundles exist. For a direct sum t = ~i of line bundlell ti oi' type 1!.

over a space X, we could define the total Chern, Pontrjagin and L­

alasses as n(1+x.), rr(1+x~) and nQ (x.), respectively, where x. = c1<e,,). 1 1 8 1 1 •

We might even hope that there is a splitting theorem analogous to

the one for ordinary complex bundles, i. e. that tllere i8 a natural

class of "bundles of type ,!!." - bundles!: such that g*,; is a sum of

line bundles,t'or some map g:Y~X for which g*:H*(X)"H*(Y) ill a

monomorphism. Then if it is also true that the classes c(g*!:), p(g*t)

and L(g*~) defined above lie in the image of go, we have definitions

for the corresponding characteristic classes for the bundle e. This

would then suggest a whole series of further questions. First, we

would want a geometric characterisation of tho~e rational homology

cobordism bundles which are of type ~ in this sense (a necessary

condition, for example, would be that the L-class is zero in dimensions

not divisible by ~s). This geometric deSCription should be such as to

allow the actual construction of the map £ with g*e a sum of line bundle~

(analogous to the well-known construction for complex bundles). We

could then ask if every bundle in our category is a direct sum of

bundles of type~, and, if so, if the representation is unique (e.g.

could it happen that a line bundle of type 6 is the direct ~um of

a line bundle of type 4 and one of type 21). This question is clearly various

related to the independence of th8~ower series Q5(t). Knowing the

answer5 to all of these questions would provide information about the

cohomology of the classifying space for rational homology cobordism

- 50 -

bundl es. For instance, if t he s plitting into bundles of type ~ always

exists and is unique, then the classifying

p~ of degree 4is for every i,s~1 (p: being ~ l-

space would contain elements

the usual ith Pontrjagin

class). The classifying space for bundles of type! and dimension ~

(i. e . having a p ull-back whi ch is a sum of E line bundles of type ~)

would be a space BUS whose cohomology conta ins the classes p~ with n ~

i~n, and there would be a map BUs .... (K(~, 2s)n (sending n line bundles n -

over a Sp<lce t o thei r direct sum) which would map p~ to the i th

elementary symmetric pOlynomial in ry1, ••• ,ry2 (where ry. is the generator t h n J

of the cohomology of the j factor K(Z.,2s » . The L-cla s5 of a

bundle with classifying map f:X ~ BUS woul d have the component n

E

in degree 4is (i~n), where L~ is a genera lized L-polynomial defined

as the multiplicative seauence in the sense of Hirzebruch with

oharaoteristic power series Qs(./f;). The first few values of the

polynomials L~ are given in the table on page 51, in the piou3 hope

tha t some day there will be a theory to ba.ck up all the se ranoie II.

(30)

- 51 -

TABLE: THE FUNCTION Qn ( t) AND RELATED POWER SERIES

() ( 1)3 (15 (17 (1)9 gn t = t + ~ t + ~)t + 7")t + §W t +

fn(t) = g~i(t) = t + {_ ~ )t3 + (_ ~ + ~ )t5 +

( 1 8 12) 7 1 10 5 5'i 5'i) 9 + - ~ + ~ - ~ t + (- §W + ~ + ~ - 45" + ~ t +

t 1 2 1 2 4- 1 6 7 6 Qn (t) = f]t) ,. 1 + ( ~ ) t + ( ¥ - '§"I")t + ( r - 15'11 + 2r ) t +

n 1 tl 4- ,6 ,0 tl

+ ( 9" - 21W - 25ff + 45IT - 81W)t +. • •

N n Q (x.)

i=1 n l.

where Pi = ai(x~, .•. ,~~) (ith elementary symmetric polynomial)

and:

- 52 -

We wish to calculate tile 1-class of X(n) = xnls , u~ing the general n formula for the 1-clas$ of a quotient space which was proved in §3. It

is clear from that formula that the calculation consilltll of three stepl!':

a calculation of the fixed-point sets (Xn)V and their equivariant normal

bundles (for all u" S ) in order to compute L'(a,Xn)E H*(Xn)V); a n

de~cription of the Gy~in homom~rphi~m .f the inclusion (xn)v C Xn in

order te evaluate L(a,Xn) E H*(Xn ); and finally a surmnation ever all at:S • n These three steps will be carried out in thi~ section and the two

follawing ones, and will result in a formula which is already in a ~hape

suitable for computations but in which the dependence of 1(X(n)) on II

has not yet been made explicit; a fourth section will then be devoted

to the tranl'>forrnation of th}!! expression into the one given in Section ti.

We begin, tben, by examining the fi:xed-point set of the action of

on Xn , The permutation ~ can be written as a product of cycles, a E S n and it ill clear from tbe defini tiDn Df tbe action on Xn that this action

has a corresponding decompo~ition as a product of actions of the form

( 1 )

Obviously for th'e standard action by cyclic permutation or' tbe fixed­r CJ

point set (X ) r consists of the points (x,. ,., x) for x X, i.e.

A C Xr (diagonal), r

(2 )

The diagonal is naturally isomorphic to X under x<-> (x, ••. ,x); this

isomorphism will be used tacitly in future, so that we shall consider

the normal bundle of the fixed-point Mt of ar as a bundle over X and

the inclueion of the fixed-point !!et a~ the diagonal map

d d: r

X

x -HI- (x, •.• , x) •

Returning to the permutatirrr £. Ott Xn, w" epecify a little more

preoilSely the de comp.:si tion into cycliC per'nutations. 1et kr be the

number of cycle~ of length £; thulS k1 i~ the number of integere i left

fixed by £. and k2 the number of pairs of integerl'> 1.1 interchanged by £.

9" The action of Sn on Xn

- 53 -

Since ~ act~ on E integer~ altogether, we have

n

i.e. we have a:!l!Iociated te the permutation 2. ef !1, •.• ,nl Ii partition

~= (k1,k2 , .•• ) of E. Clearly the number of permutationl!l with al!lseciated

partition E: il!l

N(ll-) n! (5) k 'k ' 1k l. 2ka l' 2.... • ..

since of the n! ways of putting ~ object8 into k 1+k2+ ••• numbered boxes

(there beine kr blt)xe! having r :slot:s, and the :slctt8 in each box a1:!111

being numbered), twu yield the :same permutation if and only if there

is a permutation of the kr r-boxe:s (giving the factor k r !) IIr a cyclic

permutation within an !-box (giving a factor r for each ef the kr

r-boxet».

To illustrate the s&rt of calculation which must be done when

working with the15e elements, we give another dari vation of the formula

of Macdonald for the Euler characteristic of X(n) which was proved

earHer (Prapo:!i tion 1 of §7) by a direct computation of the cohomology.

We uae the fGllowing formula fer the Euler characteri~tic ef the quotient

of a !!pace by a finite group action, which ~eems to be le~s well known

than it should be:

e{X/G) 1

TGT (6)

i.e. the de:!ired Euler characteristic is just the average over G of

the Euler characteri~tics 01' the fixed~point sets of the individUal

elements of G. [TG see that (6) holds, we work (a5 usual) with rational

coefficient5, sa that the cohomology Gf x/e is the G-invariant part of

H*(X), and use the elementary re15ult from linear algebra that the

dimension of the G-invariant part of a G-vectGr :!pace i!! the average

over G of the trace!! flJf the individual elements l!)f G. Then

e (X/G) l: (_1)i dim i:hx/G) i~O

[, (_1)i dim Hi(X)G bO

i 1 • 1 E (-1) (-IGI z tr(g"IHL(X))) '" -IGI l: e(g,X),

i~D gEG geG

- 54 -

whsre e (g, X) ill the equi variant Euler oha.racteri~t1.c

It only remaiKII t. ~h.w that

e(g,X)

thi~ i~ the Lefllchetz indsx formula, and oan alllo be immediately o~tained

froID the equivariant Atiyah-Si~ger theorem on applying it to the de lihem

cGlmplex Itf X (of. [1l] , §9, eq. (11»).J

Applying (6) to Our situation givell, !linos the fixed-paint lIet of

an element a oorrellponding to the partition (4) haa been found t. be - k~tk~+ •••

isomsrphic t. X and theref&re to have Euler characteristic e(X)k~+k2+'"

e(X(n)) )' n! ,~

rr It partition

If we ~ub~titute exprellllion (5)

00

E t n e(X(n» = n=O

N(1T) ( )k~+k2+'" eX. (e)

of' .£

for N(rr) into thia, we obtain

tk1+2k2+3k 3 + ••• e(X)ki +k2 + •••

k i !k2 ! ••• 1k12k:<."

-e(X) log (1-t) e

in agreement with equation (13) of §7. However, the method given here

has the advantage, a~ well a~ that of illustrating the technique of

manipulating average!!> over the :lymmetric group, that it oan be u:led

without Change to compute the Euler characteristic of Xn/G for any

~ubgrGup G of 3n , For example, if we take G~ An to be the Blternating

group, the o~ly change in (8) and (9) ill that the sum is restricted

to tholll! kr with kZ+kl;-+". even (5inM we only have even permutationll),

and that the factor IGI by which we hFlve to divide ir. n!/2 rather than

n! (if' n~2). We deduce immediately

FrOpo8iti~n 1: For n>1, we have the equality

(10)

We can apply exactly the same technique to the lIignature. ThUll,

- 55 -

if we ~et Tr equal to the signature of the cyclic permutation (1),

( 11 )

we see thnt the signature of a permutation a with al!50ciated partition

(4) is T~~ T~2 ••• (since signature behaves-multiplicativelY for a

product action), and a computation exactly like (9) then yields

00

t n Sign (l(n) 00

t n ( J.... Sign(a,Xn ) E E E n=O n=O

n1 aES n

00

T tr/r ] (12 ) up [ l.: r=1

r

It will foll~ from the calculation of L' (a,X n ) late r in thi~ 15ection

that the integers Tr are given by

[ T(X), e (X),

if r ill odd,

i f r i~ even,

where (X) is the signature of X, and therefore (12) becomes

Theorem 1 (Hirzebruch (14] ) : The signature of t he nth ~ymmetric product

X(n) of an even-dimensional clol!ed oriented manifold ill given by

00

l.: til Sign (X(n) ) It=0

1 e(X)j2 1 + t Sign(X)/2 (~) (~) .

Nete that the right-h/3nd lI i de af (14) is a rat ional function of t

since e(X) and 3ign(X ) are equal modulo 2.

We now turn to the main task of thi:5 section: the calculation

of the action of !!. on the normal bundle of (Xn)G in Xn, and the

re~ulting .valuation of L'(a ,Xn). Since the equivariant L-clas~ee

behave multiplicatively for product actions, we can restrict our

atte ... tio ... t. the oyclio element a <lIf' (1). As stated previGulSly, WtIl r shall identify the fixed-point set ~r with X without explicit mention,

511 that when we speak of the normal bundle tIlf the fjxed-point set we

mean its pull-back t. X under this isem&rphisffi; clearly this bundle

is isomorphic to a sum of r-1 copies of the tangent bundle of X. At

a point (x, .•• ,x) of Xr , the tangent bundle .f Xr can5illte of r-tuples

(v1 • •• ,v ) with Vi € T X, and T (~ ) consists .f r-tuples (v, ••• ,v), r x x r so we Can represent the fibre Nx tIlf the normal bundle by those vector5

with v1+ •• o+vr=O. The action .r or ill given by cyclic permutation of

- 56 -

the ve ctor~ v .• ~

Since a ha~ order r. it~ eigenvalues are e27Tik/r, where r

~ range~ from 1 to r-1. More preci~ely, for k*r/2. there i~ a subbundle

N(27Tk/r) .f N on which a act~ by the matrix r

( C0:\ 27Tk/r - ~in 21Tk/r )

lIin 21rk/r ce:\ 21Tk/r

the fibre Nx (21rk/r) af thill bundle at (x, ••• ,x) con~illtll of vector~

(v1 , ••• ,vr ) with Vj

1\0 that

N(21Tk/r) TX III TX.

( 15)

(16 )

Since the matrix (15) is equivalent to the same matrix with k replaced

by -,!!, we can 8Mume 0 < k < r/2. The remaining eigenvector -1 (in the

case that! i:\ even) has a~ eigenbundle the bundle N(1T) whe~e fibre at ~

nonsi1lt1l of vector1l (v,-v, •. .. -v), ~o

N(ll" ) TX.

The i~Gm.rphi1lm (16) refers te N(2ll"k/r) a:\ a real bundle, but it alllo

ha~ the structure of a cltmplex bundle if we require the action of or

to be given I'l:\ multiplication bye21Tik/r, and then

N(2ll"k/r) TX(iil~ • ( it»)

We now have allMmbled :lufficient inf<!)rmation to apply the Atiyah­

Singer formula, which, we recall, :ltates

e -dim Il:Ng Cc9) IT (i tan 2) ·L(yg)·L(Ng(ll"))-~.e(Ng(7T))·

0<8<11 L t (g, Y) =

IT L e (Ng ( e) ) , O<(j<1T

(19)

where LeCe) is a multiplicative ~equence defined for complex bundles e by

IT tanh io/2 tanh (x. + i872)

Xj J

the x.' 5 being formal two-dimensional cohomology Cla:lsell of the balle J th

:o;pace of e who:le k elementary symmetric polyno.:!ia.1 b ck (0. We apply (19) with y~ Xr and g; o. Fir~t consider the oa~e of

r

(20)

- 57 -

even!> Since Ng(lf) 9! TX ill then a bundle of dimenllion equal to that

of yg, the elalle (19) ie only n&n-zero in the tap dimension 2~. The

various L- and 1S-c1as~e~ all have leading term 1, and therefore can

be omitted. The Euler clallll of Ng(lf) ill then (_1)(r/2 - 1)e e(X) z,

where Z E: ~II (X) ill the c1allll of er. (1) of §. The reallon for the

eign ill that, in the Atiyah-Singer recipe, Ng(lf) ill ~upp6l1ed to be

oriellted by the natural ,wientation~ on T(Y) and on the Ng(e) for O'frr

(ae~uming that yg al~. has a given orientation, which is the calle here),

the latter being giveR the orientation co~ng from their structure all

complex bundle!. New it is known (aee for instance [nJ, p. 66) that,

if e ill an eriented bundle of dirnenllion,g, the orientation on e 8 t

:; e ... e coming frem the eemplex IItructure differll from the natural

orientation ea e~ by a factor (_1)Q(q-1)/2. ThUll fer 1 ~k((r-2)/2, the orientation en Ng(2rrk/r ) as a cemplex bundle differs from the

orillntatiol1 ef TX(/)TX by a faeter of (-" /\ and therefore the orienta­

tions on Ng(rr) given by the Atiyah-Singer procedure and by it~ natural identification with TX differ by a factor (_1)s(r-2)/2. If

we put all this information into (19), and u~e

( 21 )

we obtain

1Tk -25 5 (r/2 -1) (i tan -) • (-1 ) • e(X). z,

r (22)

or, since

( tan !!) (tan :!:(L k)) = 1, r r 2 (23 )

e(X) z € ~5(X) (r even). (24)

We now aseume that £. ill odd. Again we can us,", (21) and (23) to

see that the first prGduet in (19) simply is the. factor (this time

(_1)s(r-1)/2 ) giving the difference between the orientation of yS

from the Atiyah-Singer recipe (i.e. induced from the orientatior~ on Y

and the complex bundles Ng(&) ) and ita orientation obtained by

identifying it with X. TheI'efore (19) reduc<ls to r..:.1

2 L1(or'Xl') L(X) k~1 1 21Tk/1'(N(2lfk/r»). (25)

NIolW, by definition, the Pontrjagin clas~ of TX ill n(1 + x~) where J

- 58 -

o(TXe¢) :::: n(i - xj) (the Chern ole$~ of a bundle g@¢ ~ati~fie~ 202i+1= 0,

and we are alway~ working with rational 0r complex ooefficient~ where

tor~iQn element~ are killed). That i~, by u~ing the identification (18)

we can take fer the :let of fDrmal two-dimenl'lional cla~l\e~ in (20) the

set lXjl U [-xjJ, where the Xj are the formal two-dimen~ional cla~~e~

of X defined by

p(X) :::: n(1 +x~). J

Therefore (20) and (25) combine to give

tanh rx. J

(r-1 )/2 IT

k:::: 1

(r '1dd) ,

where in the last line we have used the trigonometric identity

(r-1 )/2 n coth (x + i~k/r)

k=-(r-1 )/2 Doth rx (r odd),

We :5tate the resulh (24) and (27) together a:\ a propo:5ition:

FrDpoaition 2: Let or be the action (1) on Xr given by cyclic

permutation of the coordinates. Then

[ e (X) z,

E r C-1'I L (X), O:5;C~s c

if r is even,

if r is odd.

(26)

(21) )

Here s :::: ~ dim X, e(X) i~ the Euler charaeteri~tic, ! i~ the element

in H2~(x) given in (1) of §8, and Le(X) is the component in ~c(x) of the L-cla~~ of X.

An immediate corollary i~ that the number

(30)

ha~ the value given in (13), completing the proof of Theorem 1.

- 59 -

§10. The ~ysin hamomerphi~m ef the diagonal ~

In this section we oompute the ~ysin homomorphism d! Qf the diagonal

d: X - xn , x ~ (Xi J ••• ,xn), (1)

which, we recall, is defined by

(2 )

where DX is the Poincare duality ieomorphiem from cohomology to homology

defined by capping with the fundamental cla~s, and d* is the map induced

by ~ in hom<llogy.

All natation will be as in previous sections; thus X is a closed

oriented manifold of even dimen~ion and two ba~es !eiJ iE1 and 1filiEI

for H*(X) are given which are dual to each other under the intersection

pairing. All that will be u~ed are elementary pr<lperties <If the various

products (cup, cap, slBnt) in homology and cohomology, as given, for

instance, in 5.6, 6.1, and 6.10 of Spanier.

We first give a formulation of the result in terme of the given

bases for H*(X); afterwards we will restate it in a basis-free manner.

Proposition 1: For x E H*(X), the following formula holds:

g. . <e. ,.,e. x, [X]> f. x ••• xf. , ~1." "~n 2::1, l.n 1.:1 l.n

where e. . = +1 is the sign obtained On rearranging e. , .• e. f. , .• f. 11_ •• In 1.1 l.n 1.1 l.n

as +e. f. , •. e. f. and taking into account graded commutativity (this II II 1 n l n

( )r(r-1V2 ~ign is equal to -1 • where .£ is the number of ei. of odd degree.)

J Proof: Since H*(Xn) is ~pannedbythe elements e. x ... xe. (j1, ... ,jnE1),

Jl In

it i3 sufficient ta ~haw that the two ~ide~ of (3) agree after we cup with

this element on the left and evaluate en [Xn]. Since <e/j,[X]> = cij '

«e. x ••• xe. Hf. >< ... xf. ), [Xn]> J1 In),' ),n

[ 0, if (i 1 , ... ,in) * (j" .•. ,jn),

if (i 1, ... , in) (j 1., ••• , j n ) • E. ., J" .. In

- 60 -

Therefore i t i~ only nec,", Mary to prove

< (e. x ••• xe. )( d, x), [xn] > " < e . ... e. x, [X J:> • J~ In' ,h In

(5 )

But from the definition of d t and elementary manipulation8 of the cup

and cap product~, we obtain

< (e. x ••• xe. )(d,x),[Xn ]> J:J. J n .

<: ". x •.• xe. , (d, x) n [xn J > Ji J n .

<: e. x •.• xe. , Dxn(d,.x):> Ji In

< e. x ... xe. , d,,(Dxx» J~ In

< d*(e. x ... xe. ), xn [Xl:> J:J. In

<d"Ce. x ... xe. )x,rX]> J;, J n

(6)

But a4«e. x ... xe. ) J~ In

e j1u ... ue jn by definition of the cup product.

We new wi~h to reformulate the content of the propo~ition in a

ba~ill-free way. To do thill, we u~e the slant product. Recall that,

fer two topological I\pace~ A and B, the ~lant product send~ an element

u E If'(AxB) and an element zE H (B) to u/z E if-q(A). If ~ ill a product q

11xb (with 11E H*(A), bE W(B», we have the formulll

(axb)/ Z <b,z:> a e: H*(A).

New take A" Xn, B" X, and z " [XJ c H2s CX). Then

(r. x .•• xf. xy)/[X] = <y,[X]:>f'. x ... xf'. E H*(Xn ) ~:J. ~n 1~ ~n

(8 )

fer all yE H*{X). Sub~tituting this into (3) and using graded commuta­

tivity, we abtain:

) 1..-.1

E. . [(fi x .•. xf. )x(e .... e. x)l/[X] 1 1 _" ·~n 1. ~n :1·1 1n

i1..7"" .,inEI

l': [(11*1f. xe. ) (1T*2 f . XI!!. ) ••• (11*f. xe. )(1xx)l/[X], ~i ~1 ~2 1., n ~n ~n

where the summation h alway~ over the same indice~. Here 1Tj i~ the

projection Xn ... X onto the }h factor, and the expre!l5ion in curly bracket!

ill lin element of H* (XnxX). In the la~t line we can bring the llummatiDn

- 61 -

in.to the product, obtaining

d,.x = f( E rr*1fixe.) ••• ( Z 1T*f.xe.)(ixx)l/[Xj. i€l ~ iEI n ~ 1

Cle8rly

where

Z iEI

and where

11":f.>< •. J 1 1

f. x e. 1 l

E H"~(XxX)

. d f .th T :L!! the prlil uct 0 the J projection and the identi'~Y map. he

(10)

(11 )

( 12)

element .!l defined by (11) is independent of the choice of balles fe i l

and {fil. One way to lIee thiiS ill to notice that, if ei = E c i / j !lnd

[i = r, dijf j are another pair of dual bases, then the matrix c is the

tranllpOlle of the inverse of ,2;, from whi ch it follows that E fixei

L: f'i xe i . Another proof ill t o notice that, by t he case n=2 of Proposi­

ti<1m 1, we have

a

\There d:X'" XxX is the diae;onal. Yet a third way b to deduce from

equation (11) the identity (for all x,y e H*(X»

< (xxy) Il, [XxX] > = < xy, [X] > (14)

and ttl notice that this characterizes ~ comp1ete1;)'. Finally, from

Lemma 6.10.1 of Spanier 'lie Bee that a iB the re5triction to XxX of

the Thorn Cla1511

whv15e existence i1'l equivalent to the orientabili ty of X.

We can now restate Proposition 1 63:

(15)

Jsi tion 2: Let d:X'" Xn be the diagonal map, Tf. :Xn ... X the projection J

) the }h factor, Tf .x 1 :XnxX-> XxX the product of Tf. with idX' and 2 J J

en H S(XxX) the element defined by (13) or (14). Then for all xe Ji*(X),

- 62 -

the image ef ! under the Gysin homomorphism d! of i is given by the

formula

(16 )

- 63 -

§11. Preliminary formula !2£ L(X(n))

We can new apply the re~ult of Chapter I, which here ~tate~

L(X(n» = (1 )

we omit the "* since we are already using it to identify H*(X(n» with

H*CXn)Sn. If £ is broken up into a product of cyclic permutations a r then L(a,Xn ) is the x-product of the corr"6ponding L(a ,Xr ). The r latter are obtained by combining the results of the preceding two

sections, and are given by

[ e(X) ZX ••• X3, ifriseven, C-I;

r !«rr1x1)*a) ••• «rr x1)*a)(1xL (x»l/[x], r c if r il:! odd.

For the first line we do not need the results of the last I:!ection,

!l i nce by definition of the GYllin homomorphil5ll and of the element Zx of the top-dimensional cohomology group of a manifold X we have

Zy = f*zX for any map f:X'" Y I'lt all.

Now if (j1, ••. ,jr) is the set of integers cyclically permuted

by IT, then the ith projection map TT.:Xr-+X il:! replaced by rr. :Xn ... X - l J.

when we identify the Xr on which O'r ac;t15 with the Xj1 " ... xxjr\m

which IT actl;(here X. denotes the jth factor of Xn). Thi.5 :5ub:5ti-- J

tution mUl;t thus be made in (2) to obtain the factor of L(a,Xn)

corresponding to the factor <7r of 5!.; for example, for even £: we

obtain e(X) (rr~ z) ... (rr* z). In principle we have to keep track J 1 J r

~f the order of the ji':s (determined up to cyclic permutation by £),

but I:!ince the oehcmology cl6.:ssea ~, band Lc(X) are all of even

degree and t hus commute with everything, we will nQt in :fact have

to do :so. Thu~, for A ~ !j1' ... ,jr l any ~ubl5et of N = !1, ••• ,nl

we have defined a clasl5 in H*(~

L(A) = [ e(X) n. A TT'!'Z, if r = IAI is even,

1.E 1.

f Il «1I .x1)"a) '{1xL (X))l/ [Xj, if r is odd, iEA l c

and fOr any IT E Sn we have

(2 )

- 64 -

n LeA) . A a cycle of a

We repeat that the class LeA) i~ even-dimensional and independent II)f

the ordering of the element~ of A.

When we calculated the Euler characteristic and signature of X (n)

in the la~t ~ection, we wrete the 5ummation over the ~ymmetric group Sn

a:s a I\Uillffiation aver all partitions.!!: of E:, followed by a :sumTIation

over the N(rr) permutations in Sn with as~ocieted partition!. In a

similar way, we now write the summation occurring in (1) a:s a :sum

GIver all part ition:s of N into subsets Ai followed by a :sum ever

all nermutaticn~ of N whose cycle:s are precillely the Ai' Since (4)

tells u~ that L(cr,Xn) then only depends on the Ai's, the latter :sum

will simply be niL (~) times the number N(A1, ",,~) sf permutations

of N conl\isting precisely of cyclic permutations of each Ai' A set

of E elernent5 clearly ha :s (r-1)1 cyclic permutation:s, ~o

'Ile a1:50 have to divide the ~Um over al l subsets A1, ... ,J\: by k!,

:linee the same partition of N into di:5joint subsets is ceunted k!

times with different numbering:5 of the Ai's, Therefore we C8n

write equation (1) 35

L(X (n) n 1:

k= 1

A1"" , ~CN .1\ ':5 ui:sjoint

A1U, ,.U~= N

1 k!

k [J (CIA I -1)! L(A.»)

j= 1 J J

(5)

(6 )

which , combined wi t h equati on (3 ) for L(A) , completes the determination

of the L-clas:s of X(n). Nevertheless, t he expre:5sion Qbtained i:s

extremely unwi eldy, and we will devate the remainder of this section and

the whole of the next one to a reformulation of it inta a better form.

We introduce dummy vari ables x1' ••• ,xn; the func t ion of x. will

be as a marker, indicating t he jth factor in the product Xn orJits

cohomology H(X)(i) ... ®H(Xn). The x j 's are supposed to commute iii th one

onotner and with the element5 of' the cohomology of Xn but to satisfy

no other relation5; i.e. we will be working in the ring H.(Xn)[x1,.",xn J of p"lynomiab in the x . ' s wi th coefficients i n H* (Xn), Ne fix t he

.J i'olloning notatiGn:l, which ,fill be used t hroughout t his and the next

- 65 -

~eotiOM: N will alvlliy/!l refer to the /!let! 1, ••• ,nJ; for ACN,

we write xA rer n x.; for f a pClwer ~erie15 or pOlynemial in iE:A ~

the xi'l\ and A It l\ub1\et of N, BAr ill the coefficient fir xA in f.

Then fer Ai'" ',1\: arbitrary lIUbMtll I9f N, the conditions that

1~k~n, that the lIetll Ai are disjoint, and that the union of the .~

ill N. are together eouivalent to the ISingle condition xAi ,,,x1\: '" ~. Theref.re (6) becomes

L(X(n» "" 1 E -

1<=0 k!

Ai" ",1\C N

xA1 ••• x~=~

oN ( Z ...!... ~ ~ «IA.I-1)! xA. L(A))) 1<=0 k! A1' •••• ~CN j=1 J J

~ (. Z (IAI-1)1 xA L(A) )\ j=1 ACN )

ON [exp ( E (I A 1-1 ) 1 x A L (A) )] • ACN

Thill exprel\lIien, equivalent to (6), ill considerably more pleal!ling. To

make further progre5l\, we must lIublltitute the value of L(A) from (3). We write th", expl!lnent i.n (7) all K(x1, ... ,xn ) or simply K, and the

c.rre5pllnd:ing sum restricted t. subset!! A with exactly E element:!!

all Kr • Thu~ for E even. we obtain

= E (r-1)1 xA e(X) zA ACN

IAI=r

(8 )

(we uae a aimilar notation till that flilt' xA' Le. zA " ,n zi' where zi JEA th

denote~ trlz = 1x .•. xzx ••• x1 E H*(Xn ), the product having a! in the i

place). Since the :lummBtion over A j:,\ B sum over unordered ~ub5etl'l

.f N, we have to divide by r! if we sum instead over all i 1, .... i r EN,

so (tl) becomes

X r

1 e (X) r

illl r

n E

i = 1 r

- 66 -

(r even).

Exactly ~imilarly, if E is odd we obtain frQffi the second line of (3):

where the exprea~ien in curly bracket~ i~ in the ring

which i:s mapped. by /rX] h the ring

More preci~ely, the expressions considered are in the 3n-invariant

subring~ R,Sn and RSn, where S act~ by simultaneously permuting the n

fecten of the ten~er product H*(Xn ) = 8~~1 H*(X) and the x i 'll. We \1rite

a = ••• + x (w x1)*a n n € R'

fer the elemenh appearing in equation~ (9) and (10). Then the

exp.~ent K of (8) i~ given by

( 11 )

( 13)

( 14)

K 00

z; "(X) zr/r 00 ~ 1

+ z; Z rC-~- ! a r (1xL (X» lICx] =1 0=0 c r=1

r eV!ln r edd

where the fUllction g 1 ill the pOl'ler Mriel'l defined in §8. TMl'l s+ -c

K is an element of the ring R defined in (12). eK therefore als8 is,

the IIymbel oN defined above map'" R t. H"(Xn), and eur final rel5ult i",

L(XCn» = E ( 16)

- 67 -

§12. ~ dep.mdence !! L(X(n)) .!!! !!

In this section we investigate the dependence en ~ ef the expression

fer LeXen)) feund in the last section, and com~lete the proof of Theorem 1

of f8, We continue to use the notations of the last section involving the

duruwy variables Xi' The elements of R = }~(xn)[X1' , •• ,xn ] which

occurred were all polynomials in elements of the form

E (fE H*(X)).

With the notation fO=z, •• "fb=1 for a ba~is of H*(X) used in the

preceding questions, the element (1) can be written all a linear

combination of the elements

t. l

(O<ii<ib) ,

sm t~at we are really only interested in the subring t[to, •.• ,tbJ ~n

of R • N~te that to = Z and tb = x 1+ ••• +xn, These ti are tne

(1 )

(2 )

ti of the theorem of §8, which wer~ there introduced 811 rather

mysterious dummy variables with the same commutation properties as

the fi' but which now are defined more naturally in terms of ordinary

scalar or commuting dummy variables x 1' ... ,xn ' To express a E R' in

terms of the t i , we use formula (11) of §10 to write

b L: ti x e i E ~[to, ... ,\]®H*(X) C RfiilH*(X) - Rt.

i=O

Then the main result (15), (16) of ~'11 is

L(X(n)

where

l J f g 1(L:t."e.)(1xL (X))/[X] 1 c=O s-c+ 1. l C

G(to' ... ,\) = (1_t~)"(X)/2 e •

"lfe now look at the effe ct of the restriction j*: H* (Xn+ 1) -+ H* (Xn) .

We label all objects in the cohomology af Xn with a 5ub~oript B' Then

it follows immediately from the description of j* (eq. (15) of §7) that

[ (ti)n '

x 1 + .•• +xl\+1'

if O~i<b,

if i"b, (6)

- 68 -

As a censequencs, wo find that, fer BCN, and any pelynemial F,

The reB~Bn that ws cannet use (7) te assert the stability under j* ef

the expre~5ien (4) is that the 5et N ittelf changes as we pass from

.+1 t. ~.

It is clear frem (6) that te ~tudy the effect ef j* en a pely­nemial F 1. to' .•• ,~ it is necessary te loek separately at the

dependence of F en the first £ variables to""'~_1 and en the last

variable tb= x1+ ••• +xn , The farmer relates to the stable elements

[no(fO)"'~_1(fb_1)J € H*(X(n)) defined in eq. (17) of ~7, namely

nO ~-1 !: aB ( to ... ~-1 )

BCN (8)

To l'Iee that (8) helds, set r = nO + •• '+~-1 and nete that the summands

en the left-hand side Qf (8) with lEI * r are certainly zero, Thus

(replacing t~O"'\~11 by t .... t. , where j1, ... ,jr E !O, .. "b-1] J 1 J r

are indices, not necessarily distinct) the left-hand side ef (8) is

L: 1~i1<,··<ir"n

i.e. we expand t .... t. as a polynemial 171 the xi' SL emIt Rl'.y term J 1 J r

in which seme x. appears te a higher power than the first, and then ~ 1

/let all the x. I s equal to one. Clearly thill yieldll ~<t. ,"" ~ ,n-r J : J 1

t. ,1 .... ,1> J r

in the nQtation ar (8) of §7, where the number ef 1'/1

ill l'I.-r. Re-exprel!~ing thi/l in termll af the natation ef (9) and (17)

ef §7 then givol5 equation (8).

It remainl5 to stUdy the effect on (4) of the dependence of G en

the variable ~. Fir!lt we nate that !linee the /!lant preduet with [X]

sends uxv te u<v,[X]> and lIince eb=z ha! the /lame dimension 095 [XJ,

the only non-zero m(Jnamials ef the ferm

(~ > 0)

are those ~ith k1="'=~_1=O, c=O, and ~=1 (for which valuel! (9)

equals ~toO). Therefore an expanlliol'l. of the exponent in (5) all a

Tayler aeriel! in pewerll .f (tbxeb ) cOnl!lil!!ta .1' the cenl!tll.nt term

(9)

- 69 -

abtained by :letting

(10)

But g' 1(t) = g {t)/t (this feH&Vf5 i mmediately fram tlle definition :;+ s

of gIl)' and the value of G(to' ""~_1'O) obtained fram (5) is exactly

the pawer series or eq. (17) af Theorem 1 in §8, sa we find that the

anly step still necessary to deduce that theorem from (4) is the

tallowing fact abeut power series.

Propositien 1: Let G = G(to' ""~-1) be 8 power series, gs the ~eries

defined in (11) at §S, and f and Q the power ~eries defined by s s

Then

Here t~ denotes the first derivative, and 1)=2 1+" '+Zn •

Praer: Ne write ~ fer x1+",+xn and ~ = ~(to) far gs(tO)/tO' The

definitien of 8A a~ the aperatar sending a power series! te the

coefficient af xA in ! is clearly equivalent ta the formula

r ( 8 f ) oXi ... oxi x1="'~~n=O

1 r

(thil$ is the reasan far the nl!'ltatiorl ()), and therefere we can make

( 11 )

(13 )

use of a generalized Leibniz t 5 rule far i) A of a product of twa functions;

!: B,CC A BJ C=A Bnc=O

Ofe apply thi:'! rule to the product on the lel't-hand side of (12) to get

(1 5 )

New we expand the exp.nential as a pewer ~erie5 and apply (1~) again:

a ( ee~ G ) = 1: o;, G N AC N . ,

( 16)

and (16) sirnplifie~ to

- 70 -

J.: i 1,···,ir EN

[ r! ,

0,

x .•.• x. , ~1 ~r

and ther~f~H'e

if IN-A-BI = r,

if IN-A-BI '" r.

Now fer any function.l:.! of one veri .. ble, and CCN, we have

0C(h(tO)) coefficient of Xc in h(x1z1+ ••• +xnzn )

= (ICI}t Zc • ooeffioient of t lCI in h(t)

Zc .(dICljdtICI )a(t)lt=o'

where Zc denotee ni~C zi' Moreover, ~ince G i~ a function only or

to' • '" \-1' and ~ince :f i Z = ° for i=O, ••• , b-1 (here we need that

( 1S)

fi i~ of po~itive degree, i.e. that G i~ independent of t b ), we

conclude that zB (JAG =< 0 if AIIB * ¢ (for if .J. i:s in both A and B,

the. in the jth place in a typical monomial we have both a factor!

and a factor fi for some i=O •••. ,b-l. and their product i~ zero).

Therefore we can substitute (19) in (18) and omit the conditien AIIB=¢:

0N(ee<p G) = E i) G E z • d lBI L· (g~(t ))l\""IAI-IB1J ACN A BCN B dt iBI -t- t=O' (20)

Comparing this with the equation (12) whioh VIe want t. prove, we :see

that it only r.main~ to preve, for O~j~n, the f.rmula

; L·· L Z ] '£L· (gl5(t))n- j- k J = Q (1))rt+1- j f'(1/). (21)

k=O BCN B dtk t 1I:S IB! .. :tc t=O

But it fellows ea~ily from z2= 0 that the fir~t expression in aquare

bracket~ in (21) i~ just ~k/k!, while the lIecGnd factor equal~

k! rest o[(g (t)/t)n-j-k t-k- 1 dt] = k! res o[Q (y)n-j+1y-k-1 r '(y)dy], '" II y= :s a

as \fe lIee by lIubstHuting Y= g~(t). t= fi'\(Y)' Equati9n (21) r.n.,,:s.

- 71 -

§13. Symmetric product~ !t spheres !E£ !t Riemann surfaces

This section centains three applicatiens ef the theerem et §8.

The first, included selely fer the purpese er illustrating the u~e

er that theerem, is • second derivatian er Hirzebruch's rarmula ter

the signature et X(n) (§9 eq. (14». The secand is an evalustien

ar the fermula ter X an 6ven-dimertsi6mal sphere; the calculation

here i~ very ea~y eince the cehAmelegy and 1-clas5 of X are trivial.

The third applicatien, rather harder, is t. the case where X is

twe-dimen!ienal, Le. a Riemann surface (the case where Q .. (t) i~ the 1\

u~ual eeries t/taAh t). In this ca~e, X(n) is III complex manifeld,

a projective fibre bundle over the Jacobian of X if n+ e (X) > 0,

and it~ Chern cla~:s hu been calculated (M8.cd~nAld (lS]). This et

course alse gives the L-c18s5 af X, and thereforo we are able ta

check the Carrectnes3 of the main thearem of this chapter.

We start, then, by evaluating Sign(X(n». Since z2 = 0, we have

(a similar fermula for all f/k/k! WIl:!! already used at the end of the

last Mctt.,lI.). Therefore

Sign(X(n» <1(X(n»,[X(n)]>

(deg ~)-i <t(X(n».~*[XnJ>

..; <11*L(X(n», [XnJ > Il..

~ c.efficient of z1'" zn in 11*1 (X(n) ~.

coefficient of TIn in 1T*t(X(n».

Frem eqs. (17) and (18) of §8 we then ebtain

Sign(X(n»

ceetficient mf fin in [f'(TI)Q (T/)n+1 x S 5

x .H(fs(f/),O, ••• ,0)/(1 _ f (f/)2)e(X)/2 J. II

The lIubsti tution t f (1) then permi t~ u~ t. rewrite this aI'\" s

(1 )

or

Sign(X (n))

L dt '" rellt _O ~

- t

QQ 11 ~ t Sign(X(n) =

11=0

- 72 -

Finally one obtai~ frem the definiticn IIf H Ceq. (16) of §8) and

the fact that 0 0 =1

H(t,O, ••• ,O) =

is zer&-dimensional that

:I

L: <g j (t eO)L (X),[X]> O s+ -c c

c=

< g1( t) (Sign(X) z), [X] >

Sign eX) g1 ( t ), (3)

( ) -1 1 1+t) which, substituted int. (2) give~ (since g1 t = tanh t = 2 log 1-t

'; t n Sign(X(n)) = (1-t~ )-e (X)/2 (~:!)Sign(X)/2, (4) n;::O

the formula which wae t. be proved.

Even :simpler ill the complete evaluatien of L (X(n» when X = S2s.

Here the basit'> it'> jU!! t eO=f 1= 1, e 1=f O=z (b= 1), and. the L-claet'> i8

of cllurse trivial. Therefore the functien H( to' ••• , t b _1) is just

1I 0:0 <gH1_c(tO"'O)Lc (X), [X] > = O.

Of ceurl5e the Euler characteril5tic ill two, sO the p.wer ~eriet'>

eH(to,.··,tb-1 )/(1_to)e(X)/2 ill juet (1-tO)-\ and theref'ere the

f!loter G ef the f0rmula fer L (X(n» b simply

f;(ry)

Therefore the whele expreesien for L(X(n» is given by

PropositieA 1: Let X = S2l!, ~ a po!itive integer, and ~ € ~s(X(n)) the u~ual element z1+",+zn' Then the L-cla~~ of X(n) is given by

(5)

- 73 -

L(X(n»

Netice that this fermula automatically has the property that

the ceeffi dent of 1)n in it ie equal tG one or till zere according as

!! i8 even er IIIdd. Thill 18 in accGrdance with (4), wince e(X) " 2

a~d Sign(X) = 0 for an even-dimensional sphere. For the Case 11=1,

X(n) is the nth symmetric preduce of P1¢' and can be naturally

identified with Pn¢; then ~ E Hf(Pn~) is the standard generater

and (6) reduces (since f 1(t) tanh t) to the usual relatigR

We will cenllider this IIpecial calle X :: S2 mere thereughly when

di~cu8l1ing the equivariant version ef our theorem.

'ife JlOW COme te the last application of thill lIection. AMume

that X is a Riemann surface IIIf genus~. For the ballis f q, O •• ,fb_1 ef the cohornelogy we choolle the standard generators of H (X), i. e.

(6)

(8 )

with Z E ~(X) as usual and the a's one-dimensional Clal5l5ell satisfying:

1lJ.0 aJo= aJ.0'rxJ~ :: 0 (all i,j), aoa'o=ll~ao=O (i'Fj),aoll i' :: -a~ao = z. (9) J.J lJ J. 11

Then the dual ballis !eil is !1,-a1, ... ,-Il~,a1, ••. ,agl. T. apply

the procedure sf §S, we must new introduce new variablell to"'" t b_1-­

here re18belled t,t1, ••• ,tg,t;, ••• ,t~ to parallel the labelling of

the fi',,--which are lIubjeot te the csmmutation rule:!!

(10)

In particular ti 2= ti2 =o. Of ovur:!e the fir:lt variable 1 commutes with

everything, since it corresPQnds to an even-dimensional cohomology

clas:!!.

Since L(X) = 1 and s=1 in our case, formula (16) of §S become~

< gd t + Il), [X] > , (11 )

where we have ulled Il t. denQte the quantity

g - E t. a ~

i=1 ~ l

is + L t!a.

i= 1 l l

- 74 -

E (12 )

Becau~e i commutes ~ith 0, we can expand the expression g2(t+S) appearing

in (11) a:s a Tayler ~eri'J5 i.n plOwer:s af O. MlirelDver, 0 is a eme­

dimensional cohemology class and [Xl h a t\,o-dime~ional homelegy

Class, so only the term g~ (t) 8"/2! ef the Taylor Mries contribute 15 , se

where to obtain the last line we have used (9), (10) and (12). We now set

and observe that the vari ous si's COJ1ll1ute with one another bui1 that

- t.t . t~t: l l J. J.

0,

2 in view of the fact that ti =0. Therefer e we obtain frem (13):

e H = eg~ (t) (S 1+"'+Sg)

g s.g"(t) n e l "

i =1

~ [ 1 + Si g~(t) + 5i2g~(t)2/2 ! + ••• 1 1=1

g rr [ 1 + 5ig ~(t)

i=1

g E ga(t)r E

t=O 1 ~i1< ..• <ir~g

WI!: nflw rec,9ll that, since 5= 1 in our case, we have

and alsG e (X) = -2g+2. Theref ore if we substitute (16) int o the

reci pe for caloulating G (§/3, eqs. (17) and (18), we .btain

1 g r 2r G = ~ech2ry (1_t~nh2ry)g- ~ g~(tanh ry) (ry/tanh ~)- x

r=O

x L [1 (a.' )1(a. ) ... 1(a~ )1(a. )] 1~i1< ••• <ir~g J. 1 ~1 ~r lr

( 13)

( 14)

(15 )

(1 6)

( 17)

- 75 -

Furthermore, we deduce from the multiplication laws (9 ) that

[1(a~ )1(a. ») ••• [1(a '. )1(0:. )]. (18) ~1 ~1 ~r ~r

We write

[1(c/)1(a.)] ~ 1

2 E H (X(co)

Then the substituticlTI into (17) of equation (18) and Gf the fact that

- tanh2 ry = sech2~ givas

g G ~ (sech ry)2g Z

r=O

~ [1 + t. tan~~D g~(tanh ry) ] ~ 1) i=1

(20)

However, frem the definition @f g~(t), it i~ clear that its derivativ~

is precisely gs_1(t)/t. M.re~ver, g1(t) = tanh-it. Therefore

d tanh- 1 t 1 tanh-it g~(t) '" at ( t ) '" t(1-e) - t2 (21)

Inserting this in (20) produces

g tanh D - 1) sech21) ) ] II [sech21] + tie rJ" •

i=1 G .. (21 )

Therefore the L-clas5 of the symmetric product X(n) is given by

1(X(n» " (........!l.....)n+1 G (-I.l...-)n-2g+1 g [ ~ tanh 1] = tanh 1) i~1 shill 1] +

¥ i ( tanh ;a~J~ I3ech 2 n ) ]. (22)

Finally, we can use the following considerations t. simplify (22).

Write

so that we have, on restricting t. H*(X(n»),

n n ~ ~ (rrk*a.)(rr1*a!)

k" 1 1= 1 ]. ~

-If i + 17 • (24 )

- 76 -

But now we can repeat an argument u~ed earlier in this section, namely

~ince the ~i 's are 3ne-dirnen~i~nal, and therefere pewer ~erie~ in 0i

b~ak erf after th~ fir~t pewer. Thus for any pewer series !,

Applying thi~ to the pewer series t/tanh t gives

to 1

tanh ~. 1

~ + siAh 1)

Sub~tituting thi~ into eq. (22) gives finally

(25)

The.rem 1 (Maodomlld [:t~); Let X be a Riemann surface af genUII So and

~ a pOllitive integer. Ghoolle a basis fer Hl(X) a5 in (9) and define

the cla:'\~ell if. ,md 1) in H2(XCn) a~ ab<lve. Then 1

L(X(n») ---1l.- n-2g+1

( tanh 1) )

g t i

i~1 tanh .ri (26)

What Macdonald in fact preyed was more. Namely, X(n) is kncwn ts

be It cemplex manifold when X i~ a Riemann surface, and Macdenald showed

that its Chern cla:,\11 iB given by

c(X(n))

We have intreduced the notation5 ai' ai. ~i' ~i' ai' 1) used abeve

in accordance with the notation of Macdonald'lI paper.

(27)

- 77 -

§14. ~ eguivariant ~

If a finite group G acts on X, then the diagonal action of G on Xn

induces ~n action on X(n) sending an unordered n-tuple Ix1 , ••• ,xnl to

the n-tuple !gox1, ••• ,goxnl. If X is an even-dimensional manifold as

in §8, th$n X(n) is a rational homology manifold and t(g,X(n») is

defined by the theory given in §4; from the result of §5 we obtain

the formula for the calculation of L(g,X(n)

1T*t (g, X(n») ~ L(goa,Xn), a€S

n

where fOr a E 3n the automorphism goa = aog of Xn i5 the mlip

sending (x 1 , ... , xn) to (goXa( 1 )' ... , goxa (n»' We therefore must

calculate L(goa,Xn) in order to evaluate the equivariant L-class

on X(n), and this Clm be done by the Atiyah-Singer theorem. The

proof is fairly similar to the proof for L(X(n» given in the last

four sections, and we will therefore give a briefer account of it,

only emphasizing the points of difference with the non-equivariant

result.

Just as before, if a is a product of two permutations, then

goa also acts on Xn as a product of the corresponding two operation5.

Therefore, since any permutation is a product of cycli.c permutations,

.,nd since the equivariant L-cll<ss is symmetric, it suffices to

consider L (goCT ,Xr ), where a is a cyclic permutation of 11, ... , r l. r r Similarly, since the equivariant Euler class e(g,X) defined in §9

(cf. equation (6) of §9) and the signature Sign(g,X) defined in §2

are multiplicative, we can express the values of these invariants

for X(n) in terms of the corresponding numbers for the cycliC

permutation or' For example, Sign(g,X(n» i.s given by a formula

exactly corresponding to (12) of §9, with the difference that Tr

now denotes Sign(goa ,xr ), and e(g,Xen) is given by a similar r

formula with T r$placed by e(goa .Xr). The values of the equivariant r r signature and Euler characteristic will be computed below for goCTr ;

we give here the re.sulting formulas for X(n):

00

1: t n e(g,X(n» n:O =1

0<> r r tt exp [ .c. e (Xg ) ),

r (2)

00

z t n Sign(g,X(n) n=O

- 78 -

00 ftr r ~ n exp L-;- e (xg ) =1

r even

The first step is to compute the fixed-point sst of goar' This

consists of the set of r-tuples (x1 , •.• ,xr ) with

2 r from which we conclude that Xi = gox2 = g oX3 = ., • = g ox1 ' Thus

( -1 1-r) 1 x,g ox, .. • ,f, ox I

In particular, the fixed-point set of goOr on Xr is isomorphic to

the fixed-point set of gr on X, As a first consequence, we obtain

fram the Lefschetz formula (sq. (7) of §9) that

and equation (2) follows. If we let 1 denote the inclusion in X

of the fixed-point set of gr, i the incluzlion in Xr of the fixed­

point set of goor' d = dr the diagonal map X-+ xr , and g the map r } (-1 1-r) from X to itself sending (x1 , ••• ,xr to xi ,g ox2, ••• ,g oXr '

and h the isoillorphism of xgr onto {xr)goar given by (id, we have

from (4) the commutative diagram

This will be used to study the inclusion! and in particular the

corresponding Gysin homomorphi sm i! = eg 0 d 0 j 0 h- 1 )! = g! d, j! ht l •

Here the Gysin homomorphisms of g and h are easy to compute since

one has in general r! = (f*)-l = (f- 1 )* for an isomorphism f:X~Y.

The Gysin homomorphism of d was calculated in §10. Finally, it r will turn out from the formula for 1'(goa ,Xr) that the Gysin

r . r homomorphism j I only needs to be applied to L I (gr, X) E all (Xg ),

on which it of' course gives L(gr,X).

Leaving till last the calculation of L'(gOqr,Xr ), we show

(6)

- 79 -

how to deduce from it ~ formul~ for L(g,XCn). We proceed ~s in the

non-equivariant case (§11). Thus for A = (j1, ••• ,jr) with the ji's

distinct elements of N = 11, ... ,nl, we define a projection map wA from Xn to Xr by

(x. , ..• , x. ) J 1 Jr

and set

(8)

The only difference with the non-equivariant situation in this part of

the calculation is that LeA) depends on the order of the ji's rather

than just on the unordered set A. To see what this dependence is, set

L' = h*L' (goa ,Xr) r r

( 9)

this is the class which will be calculated below. It depends on £, ~ and X but not on a (i.e. if we replace a by another cyclic permutation

r r of 11, ... ,r1, or equivalently renumber the! factors of xr , the value

of L; is unchanged); this will follow from the evaluation of 1; belOW

and can also be seen directly. We then set

Then by the remarks following diagram (6) we have

and therefore

L(A) =

• n r The map gowA from X to X is

( -1 -r+1) gOITA (x1, ••• ,xn ) ::: X. ,g ax. , ... ,g ox .•

J 1 J 2 Jr

Clearly, if A' = (j,(1), •.• ,jr(r») for some TESr is an r-tuple

lIame elements 1il3 A but in a different order, then

-1 -r+1 (x. , g ox. , •.• , g ox. ) J T(1) J T(2) JT(r)

(10)

(11 )

(12)

(13)

with the

- 80 -

where

Since d,L e H*(Xr ) i8 clearly symmetric, i.e. fixed under the Rction . r of Sr on H*(Xr ), we have

L (A') (gOlTA,)*(d!Lr )

1TA* g* (d,L ). T • r

( 16)

This shows the dependence of L(A') on the order of the elements of A'.

It follows that the sum of L(A') over all A' with t he same underlying

Bet .. s A is given by

t L(A') A'

where T:H"'( Xr) .. H* (Xr ) is the average of the maps g* : T

i i (g*) 1e1 x ••• x (g*) r er •

i 1,···,ir

a permutation of O,-1, ••. ,-r+1

In the non-e quivariant case, t he right-hand side of (1 7) was simply

( 17)

(18 )

r! L(A). It follows that the term (r-1 ) l L(A) occurring in (6) and

(7) of §11, and corresponding to the (r-1)! possible cyClic permuta­

tions of a set A of !: elements, must in the equi variant 5i tu.ation be

replaced by (r-1)!1Tl(T(d!Lr», and therefore that e quation (7 ) of §11

becomes

L(g,X(n) (19)

(20)

where

- 81 -

~ (r-1)! xA rrl(T(d!Lr ». ACN IAI=r

If we now substitute for d! the formula given in §10, namely

d,L ::: !«rr1x1)*a) ••. «11 x1)*a)(1xL )l/[X], . r r r

(21 )

(22 )

and proceed as in the non-equivariant case (cf. the derivation of e qs .

(10), (15) of §11), we find

K r ;:

r T( I~r (1xL ) l/[X] ),

I'

where ~ is the element defined in §11(14). r

\'fe now calculate the cla:!ses L' and L. Set 2m;: dim(Xg ). r r Proposition 1: The class L; defined in (9) is given by

l' I' [

r r e(Xg ) [xg ],

c- m (r ) <: r L~ g ,X , c;,O

if r is even,

if r is odd,

where [X~) denotes the fundamental class in cohomology of the

fixed-point set of gr on X (with the relevant twisted coefficients r r

if xg is not oriented), e(Xg ) is the Euler characteristic, and r _2c r r r

L'eg ,X) is the component in 11 (Xg ) of L'Cg ,X)€ H*(Xg ). There­c fore the class L ::: j l' € H*(X) is given by r ! I'

:::

r

[ r~m(Xg ) c~' (r X) l: r c.<-m g, ,

c;,O

if !: is even,

if r is odd,

where z is as usual the fundamental class in cohomology of X, and

Lc(gr,~) is the component in IfC(x) of L(gr,X) € H*(X).

Proof: To QPply the G-signature formula, we must calculate the

norwal bundle of xgr embedded in Xr as the fixed-point set of gO~r (i.e. embedded by the wap godoj in the notation of (6»). the

eigenvalues of the action of gog on this bundle over xgr, the r

oorresponding eigenbundles, and their characteristic cla~ses. The

idea is very simple but the details a little complicated because

of the special role played by the eigenvalue 4 in the Atiyah-Singer

formula. To wake it more clear what the splitting is, we first

(23)

(24)

(25)

- 82 -

consider the case of a complex manifold X with group action. A point

x IE X with gr ox = x i& identified under the inclusion map gdj with the -1 -r+1 pOint (x,g ox, ••. ,g ox), whioh we denote X, so

T X<s T -1 X Ql ••• Ql T -r+ 1 X x g ox g ox

(26)

w. use V to denote the vector spaoe T X, and identify T -i+1 X with V . -1 x e ox

by the map induced by g1 Then the deoomposition (26) becomes

with gour acting by

(28)

where grv1 refers to the action of gr on V which is the differential

of the map gr of X. No action of ~ appears before the vectors v2, ••• ,vr beCause we have already u~ed the map ~ in identifying (2b) and (27); the map gr then appears because we have used ~ r-1 times to identify

T -r+1 X with V ;wii therefore the map ~ frOm T -r+1 X to T -I' X V g <>x g oX g ox

gets identif'ied with the lru<p gr from V to V.

So far everything is the s~e as in the real case. However, in

the complex Case we can immediately deduce the eigenvalues of goor

from (28), namely (v1, ••• ,vr ) is an eigenvector of A€t if

where we have used A to denote the linear map gr:V-> V" and clelU'ly

this is the same as

2 Jo.. vr _1

That is, Jo.. is an eigenValue of goa exactly when Ar is an eigenvalue r

of gr lV, and the eigenspaces correspond under

r-1 v ...... (v,Jo..v, ••• ,A v)

If w. remove the eigenspace of eigenvalue 1, which corresponds to

passing from the restriction to (xr)gocrr of T(Xr ) to the normal "00 r)ge«r: bundle NQ I' of (X r, we therefore have found tha splitting

of this normal bundle (henceforth denoted N) into eigenbundles:

(29)

(30)

- 83 -

namely, for A*1 the complex sub bundle NA of N on which gour acts as

multiplication by A is given by

r

(32)

if Ar *1 (here M denotes the normal bundle of Xg in X, and Me the complex

subbundle of M on which gr acts as multiplication by Q) and by

if Ar =1. We write LA for the characteristic class L8 of §2, where

A = .iO, Le. for a complex bundl" ~ with Chern clalSs rr(1+X),

Ae2xj + LA(O IT (34-)

x A.2x j - 1 j

Then the Atiyah-Singer f'ormula together with (32), (33) gives

(35)

We now use the identity

Az+1 [ 1, if !. is even,

l=tJ :G:1 &zr+1 if r is odd.

ezr -1

w. find that if r is even, (35) reduoe s to n x ., where x. range 5 over r J J

the formal roots of the Chern class of TXg , and this product is just

the Euler class of TXgr.

(x.

IT J r tanh x.

x. (TX g ) J J

If' r ill odd,

coth rx.) J ) , and if we multiply this by rm (where m = dimtXgr) it becomes exactly

the Atiyah-Singer expression for L'(gr,X) except that each two­

dimensional class x. is replaced by rx., Le. it ill the even-J J

dimensional oohomology clalls whose component of degree 2c is r C times

the 2c-dimensional component of L'(gr,X). This proves (24) in the

complex case.

W. now turn to the real c~se, where the ide~ is similar but the

~plitting up into eigenspaces more involved. Sin~e the eigenvalues

- 84 -

of goar on N are the rth roots of the eigenvalues of gr on M (the

notations are the same as those introduced for X complex except that

Nand M are now real bundles). and since the eigenvalue -1 receives

special treatment in the Atiyah-Singer formula, we will have two cases

according as -1 is an rth root of +1 or of -1, i.e. according a8 £ is

even or odd.

For a kxk matrix A, we define an rkxrk matrix F (A) by r

o I 0 •.• 0 o 0 1 ... 0

o 0 0 ... 1 A 0 0 ... 0

where I denotes a kxk identity matrix and 0 a k><k Z\!!rO matrix. Then

(28) stat\!!s that the action of goa on T_(Xr) in terms of a basis r x

given by the isomorphism (27) is given by the m.atrix F r (A), where Ii.

is the matrix of the action of gr on T X=V. We split up the normal r x bundle M of xg in X as in §2 (eq. (26»). nam.o:ly as the direct sum

(}8)

of a real bundle Nn of dimension 2s(rr) on which gr acts as multiplica­

tion with -1 and of real bundles Ne (O<&<rr) of dimen!ion 2s(8) on

which {/ acts with respect to a suitable basis as the matrix Ae of

eq. (24) of §2 (or ra;her, as the Kronecker product Ae ®Is(e)'

Then, :since V = T (xg ) ~ M, ~nd gr acts as the identity on the x vector splice Tx(Xgr ) of dimen:sion 2m = dim Igr = 2s - 23(11") - L s(e),

O<8<1T we obtain

where (1) and (-1) denote the corresponding 1x1 matrices. We must

therefore write F (i), F (-1), and F (A~) as a direct sum of matrices r r r \I

1, -1, and Ae in order to apply the Atiyah-Singer formula to goa. r This obviously gives ( ~ denotes similarity of mat~ices)

F (1) ~ at A2rr/r 'll A4n/r \II ••• at A(r_1 )rr/r (!: odd). (40) r

Fr (1) ::: 1 ~ -1 'll ~1T/r ~ ••• ~ A(r-2)rr/r (£ even), (41 )

F (-1) ::: -1 Iii AlT/ r ~ A}rr/r + ..• til A(r-2)rr/r (£ odd), (42) r

F (-1) '" Arr/r Gl ••• ~ A(r_1) rr/r (£ even), (43) r

- d5 -

From equations (39)-(44) we obtain the eigenvalue decomposition of the

normal bundle N. Namely, if !: is ~, then

N

where

I' N '" T(Xg ) ,

71' I'

N21l1c/r !l! T(Xg )®t

Nnk/r '" M®¢: n

N !!f Me cp

If !: i5 ~. then

r-1 "'2

N E N27Tk/ r iii N7r iii k=1

where

I' N " T(Xe )0¢

27Tk/r

N '" M 1T • 7T

N1Tk/ r = M Qj~ 1T N = Me cp

r-1

E 0<8<71'

N cp

r<F'e mod 21T

(for k=1, 2, •••• ~-1),

(for k=1,3, ••. ,r-1),

(for 0<6<n, ~ ~ e (mod 2"».

r-2

E N / iii E N k.:: 1 7Tk I' 0<8<1T cp

k odd r'P=e mod 27T

r-1) (for k=1,2""'2'

(for k=1, 3, ••. ,r-2),

(for 0<8<7r, r<p .. e (mod 21T».

r We denote the Chern clall5ea of' the complex bundles T(Xg )3(1:, M~.

(45)

(46)

(47)

(4B)

(49)

(50)

(51 )

(52)

(53)

(54)

2m ( ) 23(rr)C 1T) :;(e)( 0) ~ and Me by !l. 1 1+x.. IT. 1 1+x., and n. 1 1+x., respect~vely. J= J J= J J= J

Then Bubatituting (45)-(49) in the G-signature formula (eq. (27) of §2)

gives, for!: even,

r-1 I'

• 11 L k(M ~). 11 L (Me)' k=1 K- 71' 0<8<7T cp

k odd r rqF'e mod 2"

- 86 -

Now 11T(~) = e (,;)1(0 -:I. by dafini tion for a real bundle e. If WI

substitute this in (55) we see that all the L-classes cancel (this

follows either from (36) wi th !: ~Iven or directly. if WI note that

coth(e+irr/2) = tanh e = 1/coth e, and therefore L&(e)Le+rr/ 2 (e) = 1

for any complex bundle e. so that t he L-classe3 occurring in (55) cancel in pairs), and therefore

where the first ~ denotes the Euler cill.ss, the second! the Euler

characteristic, and the square brackets the fundament al ole 5S in

cohomology. If, on the other hand, £ is ~. we get from (50)-(54-):

h*1'(goar ,xr ) = 1(Xgr). (r~~~/2 L~(T(Xgr)®It). Lrr(Mrr)

r

1'-2

k~1 L~(M1T®It) k odd r

n 1cpCMe) O<fJ<"

r<p"'6' mod 2IT

IT J 2m ( x . (r-1 )/ 2

IT coth (x. + ~») . eeM ) • j= 1 h.nh x j

2s(rr)(t&nh x~ IT J

j=1 x~ J

k=1 J r rr

r-2 n

k=1 k odd

1T i7rk») coth (Xj + ~ •

8(e) ( ) n IT IT coth(X~+¥) oeee" j=1 rcp=e mod 21T

2S~7r) tanh rxj see) e 1T IT n coth rx j'

j=1 Xj Oe8c7r j=1

2m x . e(M) IT J

1T j=1 tanh I'Xj

where the last line has been obtained by using the identity (36) for

!: odd. Clearly. if we multiply expression (58) by r m, then it is

exactly the Atiyah-Singer expression for 1 '(gr,Xr) except that each

two-dimens i onal class Xj has been multiplied by!:. This proves the

case!: odd of (24-), and therefore together with eq. (56) proves

Proposition 1.

We now return to the calculati on of L(g,X(n) . Bscause Lr is

homogeneous of top dimension for £ even (eq. (25)). we can compute

(56)

(57)

(58)

- 87 -

d,L directly, without using the ~sult of §10 on the Gysin homomorphism . r of the diagonal map~. Therefore for even £ we can replace (22) by

(£ even).

Since the aotion of G on X preserves the orientation, T(zx .•• xz)

zx ••• xz (here T is the map (H3». Therefore (21) reduces to

or, in the notation of §12, simply r

'" ~ t r K r r 0' (r even).

From (23) and (25), on the other hand, we get

(r. even),

(59)

(60)

(61 )

Kr t:;o r c-m- 1 T( fr (1xLc+s_m(gr,X» li[x]), (r odd), (62)

where m: (dim Xgr)/2 depends on £. Formulas (61) and (62), combined

with (19) and (20) and the definitions of N' ti' and 50 on as given

in §§11-12, provide a complete evaluation of the class L(g,X(n».

Just as in §12, we want to study the dependence of L(g,X(n» on n, and we know from our experience there that, to do eo, we must study the

dependence of X = K(to, ••• ,tb ) and of G = eK = G(to, .•.• \) on \,. By

eg. (61), X : K (to) for even r, so the dependenoe of K on tb only r r -comes from the odd £ terms (62). Just as in §12 (of'. 8(lB. (9), (10»,

we can write out i as E ti xei and note that, since "b = z has the same

de~e as [Xl. the only dependence of (62) on \, must come from monomials ) r-1 r

of the form (toxeo (tbxeb)LO(g IX). Since c~O and &~m, we have

c+s-m>O unless c~O and s:m. Since X is connected, xgr Can only have the

same dimension as X if it equals X, i.e. if gT = id. This is ~ further

difference from the non-equivariant case, where the L-cla~s always had

a non-1:ero leading coefficient. If gr = id, then s=m, L(gr,X) = L(X),

and LO(X) = 1. We have therefore proved that 00

=1 r odd

-5 r

f/ = id

- 88 -

If ~ has even order, then gr i~ never the identity for odd ~, 80

we find from (63) that K and hence also G are independent of~. This

does not, however, mean that L(g,X(n» ha~ especially nice stability

properties. To the contrary, L(g,X(n» and L(g,X(n+1» are completely

unrelated for! odd. For if we expand

no ~-1 G = ~ 0 to .•. t 1

TIO" '~-1 'b-(64)

(the ~um i~ taken over all nO' "'J~_1~0), we find

~ c [nO(fO)···n. -1 (fb _1)], nO" '~-1 I)

where the sum is taken over all nO""'~_1 with nu+" '+~-1 = n, and

therefore the values of dNG for ~ and fOr ~+1 involve entirely different

coeffioients of the expansion (64). If, however, the element ~ has odd order E (we assume that G acts

effectively, 50 that ~ has the same order as an element of G and as an

automorphism of X), then we obtain from (63) that

\ <f>(tO) G = G(to' ... ,\) e G(to'" "\_1,0), (66)

qI(t) =

-s p-1 p t + ••• is the power series

~ (kp)-s tkp-1 p-s gS~tP) k=1

(67)

where <pet) =

k odd

where g ( ) is the power series defined in §S. We now proceed just as 8

in the proof of Proposition 1 of §12. The proof is identical up to (20)

of §12, which in our notation states that 0NG equals

,; ~ (~ ZB)( ~ 0AG(tO,· .. ,tb_1,0») ~(<f>(t)n-j-k»t=o' J=O k=O BCN ,ACN

IBI=k IAI=j (68)

The first factor equals ~k/k!, and the second faotor is a well-defined

elem.nt G. in the etable homology group H*(X(~»). namely if we expand J

G(to'''''\_1'0) as in (64-) then Gj i5 given by expression (65) with

the ~um over all nO"" .nt _1 satisfying nO+' "+~-1 = j (a finite Imm).

The last factor can be evaluated by 8etting y = gs(tP):

dk ( , k) k ( g (tP))n-j-k dtk ~(t)n-J- = k! • coefficient of t in p-s ~

- 89 -

Clearly this is zero if n-j is not divisible by ~, while if n-j=ap it is

I _dx (p-s ~s(a»ap-kJ-k! I'esx=O Lx-a:i1' c

f' (y)dy = k! res r s 1 (p -s y)ap-k]

Y=CLfs (y)8+

k k ~p+1f'( ) k! p-asp pS • coefficient of y in y § y

Therefore (68) becomes

[nip] ~

11:0 G n-ap

pS ryap+1 f~(pSry)

fs(ps1)a+1

We have proved the following theorem:

fs(y)a+1

( 6 9)

Theorem 1: Let all notations and definitions be as in Theorem 1 of §B.

Assume further that, is an orientation-preserving diffeomorphism X'" X

of order 2. For j~O, define Gj E H*(X(oo» by

(70)

where the numbers d E ~ are the coefficients in the expansion nO' "~-1

nO ~-1 dnO ... n -1 to" '\-1

nO' • ... ~_1~O 0

= L- 00 e(Xgr ) r 00

exp ~ r to + ~ r=2 r=1

(71 )

r even r odd r

Here e(Xg ) is the Euler characteristic of the fixed-point set of gr,

Lc(gr,X) is the component in ~C(x) of the equivariant 1-class of gr,

< - ,[X]> is the IllIii.p from A to B defined in Theorem 1 of ~, and

(summation as in (18»,

where IX E A is the element of' §B( 14) and g* acts on A via its action

on H* (X). Then if l: is ~, we have

L (g,X(n»

(72)

(73)

where j* is the restriction map

- 90 -

w(X(""») -+ H"(X(n»). If E is~,

[nip] L(g,X(n) ) 1: Gn _ap Qs, p (1'))a , (74)

acoO

where Q (-) is the power series s,p

-5 P

Corollary; Let X, g be as in the theorem, dim X = 2s, ~ of .2!li! order E. Let i denote the inclusion of X(n) in X(n+p). Then

j*L(g,X(mp») Q (7) L(g,X(n). s,p n

05 )

(76)

It should perhaps be observed that the occurrence here of the

inclusion X(n) C X(n+p) rather than X(n) C X(n+1) is perfeotly

natural, since it is this map which arises in the equivariant setting.

Recall that the map j:X(n) C X(n+1) was defined by

with Xo ~ X a fixed basepoint. This map is equivariant only if

Xo is fixed by G. In general, the set xG is empty and the only way

to get an equi variant inclusion map iI! to use the map

I I p-1 I x1,·,·,xn -+ Ixo,gxo' ••• ,g xO,x1, ""xn

from X (n) to X (n+p). That is, i.nstead of mapping an unordered set

of n points of X to its union with a fixed point of X, we map it to

its union with a whole ~ GxO C X. This also means that, if we

wanted to talk about the action of G on X(oo), we really should form

E different limit spaces lim X(n), with ~ ranging over a fixed

residue olass (mod p) and inclusion maps defined by (78). These

Bpac~3 would all be homeomorphic, but would not neoessarily be the

(77)

(78)

same when considered as G-spaces. However, for our purposes (eq. (74))

we need only the cohomology of X(~), and so do not need to enter into

these subtleties.

- 91 -

§15. Eguivarilint L-classes for symmetric products of spheres

In §13 we evaluated the expressien previously obtained for L(X(n))

in two cases of especially simple nature, namely for X Ii sphere and for

X a Riemann surface. The case of a sphere was much the simpler, and is

the only one we Bre also able to cope with in the equivariant case.

Let X be Ii sphere S2s on which a finite group G acts orientably.

Since H*(X) consists only of the elements 1 and~, both preserved by G,

the group G acts trivially on the cohomology of X and of its products

and symmetric products. In particular, the averaging operator T of

§14, eq. (18) is the identity, and (using (7) of §9) we also have

e eX) 2.

Since only HOeX) and ~S(X) are non-zero, only LO(gr,X) and Lc(gr,X)

could be non-zero, and the latter equals Sign(gr,x) and is therefore

certainly zero since X has a vanishing middle homology group, The

class Lo(gr,X) is one if gr= id, and zero if gr* id. Therefore the

right-hand side of (71) of §14 is

exp [ 7 .? t r 00

-5-1 I 1 r 0 + Z r <ar,[X]

> J '1t'I" =2 r=1 - 0

r even r odd r

= id g

(1 )

(2)

since the sum over odd r. vanishes (a:: toeo' and since eO=1 has degree

zero, <ar,[X]> ~ 0) and the sum over even ~ equ~ls - log (1 - to)'

Therefore the coeffioients d of Theorem 1 of §14 (here b=1) are equal

to 1 if nO is even and to 0 ~ nO is odd. Substituting this into (70)

of §14 lind using §8(4), we find

G. [ <?, if i is odd, 0) J T)J, if i is even.

We then obtain from (73)-(75 ) of §14 that:

L(g,X(n)) [ ~, if .E is even, E: is odd, (If) Ti , if .E is evsn, E: is even,

while for odd J2.

- 92 -

n L(g,X(n)) = 1) (5 )

f ( S )Ii. S P 1)

Write q = (n/p]. The sum in (5) over O~a~q can be replaced by Ii. sum n s -a ( over -00 < Ii.<tq, since for negative!!: we have 11 fa(p 1) = 0 because

the power series fa begins with a term of positive degree, and 1)n+1 " 0

in H'"(X(n)) ). Thus it is Ii sum over a = g, q-2, q-4, ... if q"'n (mod 2)

and over a = q-1, q-3, if q t n (mad 2). Since

-c 2-1 4-q y-+y +y + ••• -q

= ~ 1-y

and similarly with ~ replaced by ~-1, we obtain from (5) that

s f'(pSry) n L(g,X(n» P 17 3 !l =

f (p 81)[Il/pJ f's(pS1) - f (pS1)2 S S

,£ odd, n .. [nip] (mad 2) ), (6)

s f~(pS1) n P 1) 1) = f (pS1)[n;p]-1 ' fg(pS17) - f (ps 17)2

S 5

,£ odd, n '"

[nip) (mod 2) ) .

We state these results as a theorem.

Theorem 1: Let X = S2s be an even-dimensi anal sphere, .!i a diffeomorphism

of X to itself of order ,£, preserving the orientation. Then the eoui­

variant L-class of the induced action of ! on the nth symmetric product

X(n) is given by equation (4), (6), or (7), depending on the values

of £, E, and n - prn/p] modulo 2. For example, if X = S2 and .l: is

odd, then

where

2 Leg, S (n»

n+1 pJ)

k [ [n/p] + 1,

[nip ],

(£ odd)

if n so [nip] (mod 2),

if n. [nip] (mOd 2).

(8 )

The last assertion follOWS since for s= 1 we have fs(t) = tanh t,

and therefore f' (t) = 1 - f (t)2 = sech2 t. The case X = S2 is of s s

interest since it provides a verification of the theorem of §14; namely,

- 93 -

the gth symmetric product of S2 is equal to complex projective spaoe Pn~' for which certain equivariant L-classes were computed in §6. To

identify S2(n) with P ¢, we write S2 as P1~' so that a point of S2 is n 2

written (2.:w), where (2o,w) E C; - 101 Imd (z:w) " (tz:tw) for t .<1;* = C - to l. For Od,n, we define

by

E (rr 2o.)(n wj ) ICN jEl J JU III=i

(10)

(11 )

CleRI'ly, if we mUltiply (z., w .) by to;¢*, then each hl.' is also multiplied J J

by ! (since the jth coordinate appears in exaotly one of the products

in each 3ummand in (11)). Therefore the point (hO: ••• :hn ) is unchanged

if we replace (z.,w.) by (tz.,tw.), so we have an induced map J J J J

h: (S2)n ... pt. (12) n

mapping (2. 1 :w1), •• ,,(2.:w) to (hO:.,.:h), where II. n n 2

for any representatives (2. .• W .) of the points in S J J

hi is given by (11)

This map is

clearly symmetric, so induces a. map

The map sending (hO: ••• :hn) € Pn(£ to the unordered n-tuple of roots

( ) f n h i n-i 0 (hi h . Zj,W j 0 the homogeneous egu .. tion Zi=O i w Z = w 0 l.S

cleRI'ly independent of the choice of h's representing the given point

of P 1:) is then an inverse to the map h, which therefore is <in isomorphism. n 1 th

Now let G = ~ C S be the gvoup of p roots of unity, acting on S2 by P

,:;o(z:w) (':;z:w)

Then G acts diagonally on 82 (n), 3ending {(z.:w.) 1 '-1 to J J J-, •••• n 2

1,0(z .:w.)l. 1 .' Using the map h to identify P t with S (n), J J J=, ••• ,n - n

we find from (14) .. nd (11) that the induced action on Pn( is

(15 )

- 94 -

In particular, the action of G on Pne is linear, and for linear actions

we calculated the equivariant L-class in §6. Substituting the definition

(15) of the action of G into the result for L(g,Pnt) given in eq. (13) of §6. we obtain

where ~ € H2 (X(n) is the usual element, and correspondg under the

isomorphism h to the Hopf class x ~ ~(P t). n

We wish to show that (16) agrees with the result of Theorem 1,

namely that L(C,S2(n) is given by (4) or by (8) according as E is

even or odd. Recall from ~6 that (16) is Ii. finite sum, the only

non-zero summands being those with A equal to one of the eigenvalues

,k. In particular, A can only contribute to (16) if it ,is Ii. pth

root of unity. Since W. want the element { to have order exactly 2. th , is Ii. primitive p root of unity and therefore A must be Ii. power

of ,. Therefore (16) can be rewritten

2 n n k-j e 2ry + 1 1«(;,S Cn)) t IT ( 71 ,

) , k-j

e2ry j=1 k=O { - 1

or "quiv"lently, since we might 105 well QBSUme , = e21Ti/ p,

n n 2 k . L(e,S (n)) ~ II ~ coth(l) + T 11'»).

j,,1 k=O

It is cleQr from either of these expressions that

2 2 L({,S (n+p) = Q L({,S (n»),

where p ,j e 2ry 1 IT (11 ~ +).

j=1 ~j e 2ry - 1 Q

This was evaluated in eq. (36) of §14

Q [ if E is even.

if 2 is odd.

( 16)

(17)

(18 )

(20)

(21 )

If 1'1" Compare this with expression (11-) for even .E or (8) for odd 2, we

see that the value of LCC.S2 (n» computed in Theorem 1 also satisfies

equation (19). Therefore if the value computed there agrees with (18)

- 95 -

for some ~, it aleo agrees for n+p, and so we only have to check values

of !! which are slIIIiller than.E' If 0" n < F, the number k defined in (9)

equals zero Or one according as !! is odd or even, so (8) states

[ n+1

P TJ ,

P7)n+1/ tanh P7],

if _n is odd (o~ <p) "n • if ,!l; is even

n+1 for odd.E' But 7] 0 in H*(Pn~)' so (22) also agree3 with the

expression (4) proved for even.E' We therefore only need to prove

that (17) and (22) are equal for O";n<p. We state this as a lemma.

Lemma 1: Let,!l;, E be integers, O~n<p, and, a primitive pth root of

unity. Then the following identity holds:

1: p

A? = 1

Proof: Define

F(z)

n )..f,k z n k=O A(k z

a rational

n n

k;:O

1 [ 1, if .!l is

+ zP + 1 = - 1 zP - 1

if !! is

function F(z) by

Then the left-hand side of (23) is the rational function

.1 ~ F(AZ). P

AP",1

G(Z)

odd,

even.

(22)

(23)

(24)

(25)

Since !. i$ BIllllller than E, the factors in (24) have t~eir ]101es in

distiact points (namely at z = s-k. k;:O,1, .•• ,n<p). and therefore F(z) th

is a rational function which has at most a simple pole for ~ a p

root of unity and is holomorphic everywhere else (including z=o).

Therefore ~(z) has the same properties. Moreover, G(z) is invariant

under z ~ ~z for any pth rODt of unity e, so ~(z) is a meromorphic

function of zp~ with a simple pole at zP=1 and regular everywhere

else. Therefore G(z) must be of the form a + b/(zP-1) for some

constants!: and 2. To evaluate these, we Observe that F(oo) = 1 and

F(O) = (_i)n, and therefore G(z) also equals 1 at z~ and ° at z=O.

It follows that G(z) is the function appearing on the right of (23). As a corollary of the lemma. the residue of G(z) at z",1 equals

2/p or 0 according as n is even or odd. Since the residue of F0z) at z=1 i~ clearly 0 fO; A*~-J (j~O,1> •.• ,n) and 2 nj*k (,k- j +1)/(,k-j-1) for A.=(-J, we obtain the identity (independent of .E)

l::~=O n~=o, k*j[C,k +(j)/(sk_(j)] 1 + ~-1 )n (26)

CHAPTER III: THE [;-SIGNATURE TREOREM

.AND SOME ELEMENTARY NUMBER THEORY

In thi s chapter we study the number-theoretical properties of

certain elementary trigonometric sums occurring in connection with

the G-signa tur e theorem. It is clear f rom the form of the G-signa ture

theorem that, if the group G i~ finite, the expression for the equi­

variant signature Sign(g,X) involves the evaluation of certain finite

sums whose terms are products of the cotangents of ra t ional multiple s

of~. The motivation for the further study of such cotan@snt sums

arose from two discoveries. One was that t he formula given by Brieskorn

for the si gnature of the variety

V a

can be expressed in terms of such a sum. The second was that cotangent

sums of this sort appear in the classical literature, and indeed in a

variety of contexts: the theory of modular functions, the Hardy­

Ramanujan-Rademacher formul a for the parti tion function, the theory of'

quadratic residues, the theory of indef i nite binary quadratic forms,

the problem of the class numbers of quadratic fields over ~, and the

problem of generating random numbers. We shall say nothing about

these classical appearances of cotangent sums (references, however,

have been given for all of them), nor--except for a brief remark about

the Legendre-Jacobi symbol and the law of quadratic reciprocity--

about their connection with the theory of group actions on manifolds.

Indeed, it still seems to be mysterious that the same expressions occur

in the theory of the transformation of the Dedekind modular ftlnction ~(z)

under the action of SL (2, Zl) and in the theory of four-dimensional

manifolds wi t h group action. The connection between the signature

theorelll on 4-manifolds and the theory of auadratic extensions of ~, on

the other hand, has been accounted for since its discovery by the WOrk

of Hir~ebruch on the Hilbert modular group and the re sol ution of certain

two-dimensional complex si ngularities; it manifests i tself, for example,

in the equality of two invariants associated to a T2-bundle over 31, one

- 97 -

defined topologically and the other purely number-theoretic.

To see the sort of topological expression which arises. we special­

ize the G-si~ature theorem to the case of a component of a fixed-point

set Xl that reduces to a simgle point~. Thus x is an isolated fixed . f -', = e2i6j pOlnt 0 ~ and the action of ~ on TxX is given by eigenvalues A J

*1 (j=1, ... ,n, 2n= dim X), and the G-signature theorem gives

(2 )

as the contribution to Sign(g,X) from the component lxl of Xg • If ~ has order~, then $j must be a multiple Sj of rr/p, where a j is an

integer defined modulo £; if further the pOint ~ is an isolated fixed

point of all the powers of £. (except of course of gP", 1), the integers

a j (j=1, •••• n) muet all be mutually prime to E. Then the contribution

of Ixl to the sum EheG Sign(h,X) (where G is the cyclic group generated

by ~) is equal to

p-1 Z

k=1

(C'= e"'fi/r)

p-1 i -n Z t Hka1 cot rrkan co p ... P

k=1

The reason for the interest in the expression E Sign(h,X) is its

(3)

appearance in the formula for the si~ature of the qllotient X/G (§3 (1».

For any submanifold Y of X, we define defy (the "signature defect of Y";

aee Hirzebruch [17]) as the sum, taken over all gEG for which Y is a

component of Xg, of the Atiyah-Singer expression for the contribution

from Y to Sign(g,X). This is of course only non-zero for finitely

many lIlllnifolds Y, which are necessarily connected and of even codimen­

sian in X. Then

IGI Sign(X/G) ); Sign(g,X) gEG

Sign X + " defy nx

50 the numbers defy can be thought of as defects specifying the

amount by which the formula Sign X = IGI Sign X/G (which holds for

free actions) fails to be true. In the special case that G is cycliC

and 2!: an isolated fixed-point of every g € G -11] , we obtain (3).

- 9t3 -

Notice that the expression def(p;Q1, ••. ,Qn) defined in 0) is a

rational number. Indeed, in the second to the last formula of (3),

we could have taken any primitive pth root of unity fOr ( without

changing the sum, 50 the sum belongs to the subfield of the cyclotOmiC

field of order .E of elements equal to all their conjugates, and this '\+1 p sub field is precisely~. Moreover, if c - A-1 where A = 1, then

(c+1)P :: (c-1)P, so.!? satisfies an algebraic equation wHh integer

coefficients and leading coefficient.E, and therefore pc is an

algebraic integer. It follows that the expression (3) when multi­

plied by pn is a rational integer:

pn def(p;a1 , ••• ,an ) E ~

n n-1 ",e will show in §16 that the p in (5) can be replaced by p or

by Pn ' where ~n is a known integer and is independent of .E (it is

the denominator of the Hirzebruch L -polynoudal: see (11)). The n

same sort of reasoning used to prove (5) shows that defy is always 1 a rational number, and indeed an element of' :zed]' where d", IGI.

These remarks give some idea of the subject matter of this

chapter and of the type of interplay which takes place between the

topological and the number-theoretical aspects of expressions such

as def(p;a1, ... ,an)' A more precise description of the contents of

the chapter is as follows: The purely number-theoretic aspects of

(5)

the cotangent sums are considered in §16. We first give an elementary

trea tment of the exore ssion (3) for n=2 (this is the sum the t appears

most in the 11 terature, in connection with modular functions and

quadratic fields). shOWing its connection with the Legendre-Jacobi

symbol and giv:i.ng an elementary proof of a reciprocity law, due to

Rademacher, from which the law of quadratio reciprooity can be

deduced. We then consider the general case, giving a rational expres­

sion for def(p;81 ••• "sn) and proving a generalization of the

Rademacher reoiprocity law; the latter is then used to prove the

above-mentioned result on the denominator of def(p;s1'" .,an). The

formula giving the signature of the Brieskorn variety as a cotangent

sum is also proved. In §17 we construct two explicit group actions

for which the statement of the G-signature theorem reduces to the

Rademacher reciprocity law (or rather its generalization to higher E)' One of these is precisely the aotion of a produot of cyclic groups

- 99 -

on Pn~ (the Bott action) which was studied in §6. The other is a direct

generaliza tion of the COfl8truction given by Hirzebruch [17] for the

case n=2, but is included because it requires the theorem of §3 for

the L-class of a quotient space and thus provides a further link with

the results of Chapter 1. In the final section, we give a computation

of Sign(g,V ). where V is the manifold (1) and ~ acts by multiplying a a

zk with an ~ th root of unity. The computation is exactly parallel to

Brieskorn's in the Case g=id. The result can be rewritten as a

trigonometric sum by using the results of §16, the case g=id being

the formula for Sign(V ) mentioned at the beginning of the introduction. a

That this formula involves cotangents suggests that it can be obtained

by the use of the G-signature theorem, and indeed this can be done:

one studies a certain hypersurfaoe in Pn~ invariant under the Bott

action. However, this alternate proof will not be given here; it was

gi ven by Hirzebruch in the oourse of a series of lectures in which

a direot proof was given of the result of Batt proved in §6 ["11]. In

one special oase, we do give an evaluation using the G-signature

thecrem. Namely, when the aotion of G on Va i s free, we can oalculate

Sign(g,V ) by replacing the non-compaot manifold V with the bounded a 2n a 2

manifold Va nD ,and if we then glue onto the boundary t he D -bundle

associated to the 3 1-bundle (V n S2n-1) ... (V n S2n-1 )/31• we obtain a a

a olosed G-manifold to whioh the G-signature theorem can be applied.

- 100 -

§16. Elementary properties or cotangent ~

We wish to study the trigonometrioal SUill5

def(p;a~, ••. ,an) = (_1)n/2 p-1

Z k=1

nka1 cot nkan cot p ... p

derined in the introduction, where E is a positive integer and the

Ne can write (-1 )n/2 for i -n in (1)

for odd;! (the substitution k->p-k

ai's are integers prime to E. since the sum is clearly zero

replaoes eaoh cotangent by its negative).

We be gin by an !Ilementary treatment of def (p; <1, r). We shall

later give a rational expression for (1) which simplifies, when n=2, to

def(pj<1,r) p

4p 1: «k9.)) ((~), k=1 P P

(2 )

where (ex» is the standard notation

( (x) ) [x] - 2' if ~ is not an integer,

0) 0, if x is an integer.

w. can always take r=1 in (2),

runs over all residues (mOd 1')

since !: is

as k do .. s.

prime

Then

to E and therefore kr

(2) can be rewritten

def(pjq,l)

where

-6p

-

p Z

k=1

2 (q,P)D ' '3

(5 )

This last quantity is the "Dedekind symbol," studied by Dedekind [5]

in connection with the behaviour of his modular function ry(z) under

the action of modular transformatiolll! (he used the notation (q, p), but

we will reserve this notation for its usual meaning as the greatest

common divisor of two integers .9. and E)' The rea.son f or the factor 6p

in (5) is that it is exactly the factor required to make (q,P)D an

integer.

To study the expression (5), we introduce a slightly more

convenient integer-valued function, namely

- 101 -

f(p,g) = p-1

l: k[!sg]. k=1 p

(6)

We also note the formula (valid if (p,q)=1, as will always be assumed)

p-1 Z [S]

k:=1 P

(p-1 ) (9-1 ) 2

This can be obtained by counting the lattice points below the diagonal

of the rectangle O<x<p, O<y<q, or by substituting p-k for ~ in (6).

Since (p, q) = 1, the set of' numbers qk - p[.9!:] runs over a complete p

set of non-~ero residues (mOd p) as ~ does, so

or, evaluating the various terms, p-1

(q2 -1)(p-1)(2p-1) - 12 q f(p,q) + 6 p Z [~J~ k=1 p

On the other hand,

feq, p) =

Z O<k<p

E O<x<q

(2g-1)(q-1)(p-1) 4

Combining (8) and (9), we obtain

Ex) O<Mxp/q

p-1 1 z [.9£]~. 2 k=1 P

q f(p,q) + p f(q,p) = 1~ (p-1)(q-1)(8pq-p-q-1).

O.

On the other hand, it is easy to write f(p, q) in tnrms of (q, p lD 1 1

f(p,q) = 12 (p-1)(4pq - 2q - 3p) - 6 (q,P)D

(notice that it follows immediately from this formula that (q,P)D is

an integer). Therefore we can write (10) in terms of (q,P)D;

p (p,q)n + q (q,p)n £2 + 92 + 1 - 3P9 2

(8 )

(9)

(10)

(10A)

(11 )

- 102 -

Equation (11) is the reciprocity law of Dedekind. Together with the

fact that (q,p)p only depends on the residue class of S mod~, it

suffices to define the symbol (q,p)p entirely (by a Euclidean algorithm).

Stated in terms of the defect, eq. (11) becomes

1 - def(p;q,1) p

1 + - def(qjp,1}

q _ £2 + 92 + 1

3pq

This is the form in which the reciprocity law Was generalized by

Rademacher [.36J, who proved that

1 1 1 iidef(p;q,r) + g:def(q;p,r) + z:def(r;p,q)

where p, g,r are positive and mutually prime. We will prove later a

generali~ation of (13) for the higher defect symbols defined in (1).

( 12)

( 13)

First, however, we complete the "elementary" part of this section

by relating the Dedekind symbol (q,P)D and the Legendre-Jacobi symbol

(~). This symbol is defined whenever q,p are relatively prime integers

with p odd (e.g. as the sign of the permutation on ~/p:iZ induced by

multiplioation with s), and is given, just as in the speoial case

of prime p. by G-aui~' s lemma, namely

where

N q,p

N (-1) q, p

IIx: 1"X"~,

Therefore, modulo 2, we have

N q,p :; Z o <x < p/2

[2qx/pJ odd

p-1

(p odd, (q, p) 1 ),

qx - p[~J > ¥ l I •

(p-1 )/2 ~ [~J

p x=1

p-1

(14)

(15 )

Z k=1

k even

Z (k-1) [~] lc: 1 P

:f• ( ) _ (p-1)(o-1) P, q 2·

From (14) we deduce

(mod 4),

and combining equations (lOA) (16), and (17), we obtain1he desired

relation· between the Dedekind and Legendre-Jacobi ~ymbols;

• This was known to De de kind (Crelle 83 (1877) 262-292; Gesammelte Werke, Band I, 174-201, §6). --

(16 )

E..!..2. 2

- 103 -

(mod 4). (18 )

Sina. (~) can only take on the values t1, it is completely determined

by (18), so ihat a knowledge of the Dedeldnd symbol also gives the

Legendre-Jacobi symbol. Also, if we substitute (18) into the Dedekind

reoiproCli ty law (11), we obtain, after a short cal culation,

(s) + (12) '" (p-1)(g-1) + 2 (mod 4), p q 2

p,q odd, (p,q)",1,

which is precisely the law of quadratic reciprooity.

We now turn to the general case of definition (1). Our first

goal is a rational expression for def(p;81, ••• ,an) generalizing

equation (2). This is given by

Theorem 1: Let £ be 8 positive integer, and ~j (j=1 ••••• n) integers

prime to £. Then

z ( (!s.) ) • • • (( !.D. )) • 1~k~, •.• ,kn"P p p

pi 81kl + •• '+8nkn

.E.r.2.2!: We will make frequent use of the well-known dual formulas

z )/ [ 0, if P1' r, i\P ",1 p, if P I r,

p )..k [ 0, if A * 1, !:

k=1 p, ifil.=1.

( 19)

(20)

(21 )

where in the latter formula A is any pth root of unity. Our proof of

eg. (19) will be based on the following formula of Eisenstein:

« ~ » iI. 1 1 p-1 21Tka k

A-a ....!.... c =- l': sin -- oot !!...-. A-1 2p k=1 p P

(22 )

This Can be proved in several ways; the eaBiest 1s to note that the

differenoe of the right-hand side for a = b and a= b -1 is (using (20»)

~ [

2p'

1jp

=1 2p

if b .. 0,1 (mod p)

if b * 0,1 (mod p)

- 104 -

From (20) we obtain the formula A. a + 1 '\1 +1 1 P

,\~k Aak [ "a _ 1 if ,\*1,

l:: Z

Af=1 '\1-1 P k:1 0, if A",1.

A,-*1

th ( ) where A is any p root of unity. Substituting this into 1 gives

.J.. n

P

A8n + 1 ... ---

AP :A.f= •• • =A.~= 1 1 ~kl' ••• , kn"p Ai, ••• ,An*1

A8n -1

The sum over A can now be evaluated using (20), and the sum over A.j

(j=1, •.• ,n) using (22). The result is precisely (19).

Since the denominator of «~») is at most 2p, it follows from p

the theorem that

This is a sharpening of eq. (5) of the introduction, and will be

further improved later.

The second theorem we state for the numbers def(p;!l1,. .. ,an)

is a generalization of the Rademacher reciprocity law (12).

(23)

Theorem 2: Let aO, ••• ,82k be positive and mutually prime integers.

Then

2k 1:

j= 0

Here a j denotes the omission of a j , and Lk (P1' ...• Pk) i8 the kth

Hirzebruch L-polynomial in the variables Pj = OJ(a~, ••• ,a2k), where

o. is the jth elementary symmetric polynOmial. J

(24)

- 105 -

~: Consider the poles of the rational function

2k a' f(z) IT

Z J + 1 (25) 2z j=O zaj - 1

TheslI clearly lie at 0,00, and those points ~ on the unit circle with a'

z J 1 for SODIII 1. Since the a. 's are prime to one another, it can J

a· ai only happen that z z J = 1 if .! itself is 1 ; therefore the pole

a' of f at a pOint.! with z J=1, z*1 is simple and its residue can be

calculated immediately from (25). Applying the residue theorem gives

o L res t ( i' (z) dz ) tEt Z=

2k resz=O + resz=oo + l:

j=O E resz=t + res z=1

taj ='1

M1

-1 .:1. 2k 1 2k t13.j + - + + ~ n + resz=1 (r(z) dz).

2 2 j=O aj 1=0 taj -

i*j

Finally, we find on substituting e2t for .! that

( 2t rest=O f (e )

rest=O coth aot .,. eoth a2kt dt

rest=O L 2k ajt dt ] IT

t 2k+1 aO···a2k j=O tanh ajt

By definition of the L-polynomials, however,

2k a.t IT J

tanh a.t j=O J

OQ 2 ~ L (P1""'P ) t r.

O rr r=

Therefore

resz=t (r(z) dz) Lk (P1' ... , Pk)

a O" .a2k

and the proD!' of the theorem is complete.

The easel k=1 of Theorem 2 is just eg, (12). For k=2, the

right-hand side of (24) is

(26)

- 106 -

where s,b,c,d,e are positive and mutually prime integers and

Just all the numbers def(pjq,r) turned out to be multiples of 1/3 (eq. (4)),

we can deduce from the reciprocity law for k=2 that 45 def(a;b,c,d,e)

is an integer. In general, let ~k be the denominator of Lk, i.e. the

smallest positive integer such that ~kLk(P1, •••• Pk) is a pOlynomial

with integer, coefficients. Thus 11 1=3. /102=45, and in general (see [11])

[;~1 ] ~k '" n 1

where 1 runs over all odd primes. If we multiply eouation (24) by

11k aO" .a2k, the right-hand side is oertainly an integer. Because

(27)

the aj's are relatively prime, and because def(aj;sO' ••• ,a j , .•• ,a2k)

is in 7Z[..1.) (byeq. (23)), we can deduce from this that a j

This is only true by the above argument if all of the 8 j 'S are prime

to one another. However, to see that iJk def ( Na:!., ••• , an) is alwlEYs

an integer (here n .. 2k) , we observe that its value only depends on

(28)

the residue classes of ai (mad p), and that ai is prime to~; therefore

we can use Dirichlet's theorem to replace each a j by a large prime

without changing its class in 7Z/p~, and so we oan assume that the aj's

are prime to one another as well as to~. Combining this with (23)

and the formula (27) for ~k' we obt~in:

Theorem 3: Let p~1 and a1, •.• ,an (n"'2k) be integers with (a.,p)",1 for J

all.J.. Then der(pjB1,' •• ,an) is a rational number whose denominator

divides ~k (independently of £ and the ai's). More preCisely, the

denominator is at most equal to

[1~1 ] n 1 t

where the product runs over the odd prime divisors 1 of £. We note that Theorem} is quite sharp. If £ is prime to ~k' of

course, it states that def(p;a;., ... ,a n) is an integer (this 1s the

case if £ has no prime factor :;;n+11 but if' £ and ~k do have a common

factor, then the bound f'or the denominator in (29) really can ocour.

(29)

- 107 -

For n=2, for instance, def(p;~,r) is an integer if and only if £ is

not a multiple of 3 (this Can be seen easily from the treatment of

(q,P)D given at the beginning of the section). For higher !2. Wt'l

can test the sharpness of (29) by taking E small. Thus

2 . 2k 1 2k 1 2k .!: (cot 1) = 0f5) + (5'3) J=1

def(3; 1, ... ,1)

k has denominator exactly 3 , and

4 . 2k def(5;1, ••• ,1) = E(cot~)

j=o1

i [ (5 + 2:l5)k + (5 - 2-I5)k ] 5

2

?

has denominator at most 5(k/2] (exactly 5(k/2] unless ~ is an odd

multiple of 5).

The laft trigonometric sum evaluated in this seetion will be the

one giving the signature of the Brieskorn manifold Va (eo. (1) of the

introduction to the ohapter). It diff_rs from the previously

considered sums in that the expression given for it will be an

integer rather than just a rational number. The reason is that the

sum only involves cotangents of the form cot ¥ U odd). Just as

we observed that c = ~~~ is 1/p times an alge~raic integer if A is

a pth root of unity (cf. the discussion leading to (3) of the intro­

duction), we see that .£ oitself is an algebraic integer if iI.P= -1,

since then the equation (C+1)P + (c-1)P = 0 leads to an algebraic

equation for .£ with leading coefficient 1. Thus cot ~ (.1 odd) is

an algebraic integer, and a rational expression involving such

cotangents is therefore 1i 1'Hiional integer. We now state

Theorem 4: Let a~; ••• ,an ~ 2 be integers. and

!( ) ~ ""n •• O<x.<a. (. 1 ) x~, ••• , Xn - "" J J J=, ••• ,n ,

- I 1 x E Zln: 0 < x. < a., 1 < ~ + J J a1

+ !.o. < 1 (mOd 2) 1 I an

+ ~ < 2 (mod 2) l I an

(30)

- 108 -

be the expression gi~n by Brieskorn [3J for the signature of the

variety Va' Here O<y<1 (mod 2) means that 2k<y<2k+1 for some

integer k. Then t (a) :;: t(a1' •.. , an) is zerO if n is even, and 11-1

tea) (_1)2

N

2N - 1 1:

j=1 j odd

t..i!!: t..i..!L oot k 00 2N 00 2a1' • • 2an

if Q is odd. where N is any multiple of the integers ai, •••• a n•

~: The statement for even E follows immediately on replacing

each Xj by aj-x j in (30). He could prove (31) by one of the methods

used earlier in the seotion (e.g. by a residue or Fourier series

technique or using the formula of Eisenstein), but prefer the

following Illore elementary approaoh. Let N> 0 be as in the theorem

and set b. = N/a. (j=1 •••• ,n). Define a polynomial f(t) by J J

ret) l: O<X:l<ai

II. b. 2b· (aj-1)b j n td+t J T ... + t ) j= 1

n n

j=1

b· t J

N t

6· t J

If we write c for the coefficient of t r in f( t). it is clear that r

the number t(al, ... ,an) is precisely c1+ .. '+~-1- cN+1-·· .-c2N- 1 +

c2N+1 + ••• + c 3N- 1 - ••• , i.e.

t(a)

where

The residue in (34) is well-defined since f(t) is a polynomial and

f(t-i) is therefore meromorphic at t=O. Since ret) is a polynomial,

moreover, f(t- l ) has no other pOle, and it is clear from (35) that

get) has poles only at points t with t N = -1, these being simple. The

- 109 -

residue theorem therefore gives

t (a) b fez) ~ N

res t : z [ t ]

z '" -1

n b· t J + 1 .Lt...1 -1

~ n tbj

t N" -1 j,,1 1 - 1 - t N

This is e qui valent to equation (1).

Notice that the theorem gives even more information sbout the

cotangent sum appearing than the remarks preceding t he theorem, for

from these remarks it only follows that a cotangent sum involving

terms cot ~ must be an integer, while it follows from (31) that 28.

the cotangent sum a ppearing there is actually a multiple of N.

It is possibl e to prove a large number of similar results. For

example, a specialiZation of a very slight generalization of Theorem 4 (see Hirzebruch [leJ) yields an integer expression for the quantity

2p - 1 b

j:1 j odd

~ ~ cot 2p •.. cot 2p

der (p; 8.1, ••. ,a n) ] ,

where £ is a positive integer and the ai's are odd int egers prime

to~. This expression gives a formula for the Browder-Livesay of

the free involution T on the lens space t(p;S1, ••• ,sn) defined as

the covering translation of the double covering

However, we have only given in this section the formulas relating

to the topological situations considered in §§17-18.

- 110 -

§17. Group actions ~ Rademacher reciprocity

In this section we construct two finite group actions on complex

manifolds for which the equality of the G-signature theorem is

precisely the generalized Rademacher reciprocity law proved in §16. To obtain the defect~ def(p;a 1 , ••• ,an ), we have to look at manifolds

of complex dimension ~; thus the original reciprocity law of Rademacher

(eq. (13) of §16) corresponds to manifolds of real dimension four.

The first action is the same as that studied in §6, namely the

linear action of G ~"' x ••• x f.I on Pn¢, where f.la denotes the th 8 0 an

group of a roots of unity and aO' ••• ,an are mutually prime integer ••

The calculation is relatively short because the fixed-point sets and

their normal bundles were already found in §6.

The other action considered is the action of a finite cyclic

group on a space obtained as the quotient of another finite group

action on a hypersurface in complex projective space. This situation

was used by Hirzebruch [17] to obtain the classical Rad~macher recipro­

city law from the G-signature theorem; the only difference is that for

manifolds of dimension higher than four we need the whole L-class and

not just the signature of certain quotient spaces, and therefore the

use of the G-signature theorem must be replaced by the use of the

result of §3 of Chapter I.

(I) ~ group ~ 2:! projective space

We will use without comment the notations of §6, thus X ~ Pn¢,

~ = ~ x ••• x~ , and X(C) denotes the component of Xg (for a fixed g€G) a o an

given in §6(5). It follows from the assumption that the a .'s are J

mutually relatively prime that each Xes) is empty or consists of

exactly one point if s~1: it is empty if ( is different from all of

the coordinates (i of ~,and consists of the single point

(0: ••. :0:1:0: ••. ;0) (ith coordinate = 1) ( 1 )

if (= (i " 1. We write Sign(i) for the total contribution of the

point Pi to E Sign(g,X), i.e. g€G

Sign(i) (i=O, ... ,n). (2)

- 111 -

n Thus E Sign(i) represents the total oDntribution to Z Sign(g,X) from

~o ~G

all fixed-point set components xC!:) with 1:*1. Similarly, w. write S for

the total contribution from the components X(1), i.e.

Then

S = L < L'(g,X)1,[X(1)]> gE:G

n S + E Sign(i) = Z Sign(g,X)

i=O g€G I GI Sign X/G,

by the usual formula for the signature of a quotient. Moreover, the

action of G embede in an action of the connected group ~+1 (cf. §6),

so G acts trivially on ft*(X) and therefore Sign xjG = Sign X

Sign Pnt. We can aSllume that !l is even (otherwise all the signatures

are zero), so that Sign Pnf, = 1. Since IGI = aO' •• an , we obtain

n S + L Sign(i) = aO .•• an

i=O

The Rademaoher reciprocity law will thus have been exhibited as a

special case Df the G-signature formula when we have shown that:

Sign(i) 8 0 , .. 8. .... a defCa.;aO'" .,a., ... ,8 ), ~ n ~ ~ n

(6)

s re s 0 [~ a J dy ] • (7 ) y= j=O tanh a.jY

In the last equation we have used the notation of §16; thus k = n/2

and Pj = aj(a~, ••• ,a~k)'

To prove (6), w. consider gEG, {={i*1. The eigenvalues of £ on ( f 1<.6) ,.-:1.(. the tangent space at the isolated fixed point Pi are C.">i !> J

and therefore (2) becomes

j~O (

1: :l.{. +

) Sign(i) E ~ J ;; i(

C~i 1 c~j = 1 j

~ j:H 'J (i",1

- 112 -

j~O (

-a·

t a. S;i. ~ + ) (8)

J -a· • ;::'1- '" 1 (i ~ -

. ~ j*i (i'" 1

where the last line has been obtained by a simple trigonometric

identity: The right-hand side of (8) is preoisely the expression (6).

To compute S. we need to evaluate L'(g,X)l on [X(1)]. Renumber

the coordinates (for a fixed g = ('0' '''''n) E G) so that '0="'='5 .. 1, ~i*1 (i=s+1, .•• ,n). Then X(1) = P3~' and if we denote the generator

of H2(Fs~) by ~, we Can express L'(g,X)1 by

8+1 n ,.e2y + L'(g,X)1 (~) n ~2y (9)

i=5+1 Sie -

(eq. (10) of §6). Evaluation on [XCi)] corresponds to taking the

coefficient of yS in (9), so we obtain

as the total contribution from Ps~ the expression

( ) s+1 n ( ) ] = res 0 ( ooth y II a. coth aiy - ooth y dy , y= i=8+1 l

where again the last line has been obtained by using an elementary

identity. The unsymmetric form of (11) is due to the renumbering of

the coordinates. In general, if I is a subset of N= fO.1, •••• nl.

(10 )

(11 )

then the total contribution to S from elements gEG with (i~1 ~ iE1 is

resy=O (ITid(cothy) nine aicothaiy - cothy)dy].

* Namely §6(17).

- 113 -

To obtain S. we must sum this over all subsets I of N. getting

3 res 0 [ ~ r leN

n (oath y) iEI

n (a.ootha,y - eoth y) i¢I ~ ..

n (coth y -;- ai eoth aiy - coth y) dy l. ieN

which is equal to the right-hand side of equation (7).

(II) A ~ ~ £ll ~ hyper surface

dy ]

We change our notation somewhat. We now let!! be an odd

integer and b~, ... ,b n mutually relatively prime integers (all positive).

We write N for the product b 1 ••• b n and ai for the quotient N/b i • Let

G- denote the group IlN and H the group !ltd' •.. xl1J n • The product GxH

acts on the hyper surf ace

v =

by

N Zo + ••• + N

z :: 0 J n

(12 )

N bi where \: =1 and (li =1. It is easy to see that the map Y->Pn-1' given

by projection onto the last n coordinates gives an isomorphism from

V/G onto P 1" The induced action of R on P 1¢ is of the type n- n-considered above and in §6 and acts trivially on the cohomology, so

Sign(V/GxH) = Sign(P 1C/H) = Sign P 1C = 1, n- n-

the last equality hal ding because of the assumption that n is odd.

( 11+)

The quotient W = viR is naturally a complex manifold, sinoe

the aotion of H on V, although it does have fixed pOints, has at a

fixed point a representation which is a sum of the standard represen­

tation of J.lb on (:, and the quotient of this standard repre5entation

is non-singular (the map z ~ zb gives an isomorphism C/J.lb ~ ~). We

will apply the G-signature theorem to the action of G on W, using

Sign(W/G) Sign (V/CxH) 1 •

L.t Y be the hypersurfaoe in W defined by

(16 )

- 114 -

Since G aots trivially on Y, we have YCWg for every g€G. W. claim

that, for gE G - /1l, the fixed-point set w€ is the disjoint union of

Y and finitely many isolated points. Indeed, we can identify W-Y

with the Brieskorn variety

b· by the map wi: (Zi!ZO) 2, and in terms of these coordinates Wj for

W-Y, the action of G is given by

Wi., .... , b n ) {; Wn.

( 17)

( 18)

b. Therefore w is a fixed point of !: if and only if (1: 1 whenever w. ,*,0.

1

This can be the case for at most one i if S is not the identity (since

the b. 's are relat.ively prime), and then the condition that w lies 1 a'

on the variety (17) is w. 1 = -1. Thu.s s acts freely on W-Y if each b· 1 b·

I: J is different from one, while if ( 1= 1, 1:*1, the fixed-point set

of C on W-Y is the set of a i points

Wei) 1(0, ... ,wi'" .,0) a.

w. 1 = -1 lew. 1

From this description of the action of G on W, we obtain

N [GI IGI Sign wiG

Sign W + n

~ Sign(';:,W) ~EG

~ S., i=1 1

where Sy is the sum Over S € G - [1l of the contribution to SignC!:, W)

of the component Y of W , and 3i is similarly the total contribution

from the set Wei). IVe claim that

Sign W =

a. def(b.; b 1 , ••• , b., ... , b ), 1 1 1 n

n resv:o [(N - coth y tanh Ny) TI (coth b.y) dy].

" j=1 J

res 0 [(ooth Y y=

n tanh Ny) IT (oath b.y) dy].

j=1 J

Clearly the Rademacher law follows from equations (19)-(22).

We begin by calculating Si' If WE 'N(i) is an isolated fixed

point of !:. then we see from (11:1) that the eigenvalues of !: at '/IT are

(20)

(21 )

(22)

- 115 -

b· the numbers (J (1~j~n, j*i), and the contribution of ! to Sign«(,W)

is therefore equal to

n b·

n ~ J + 1

j=1 ,b j _

j'l.i

We must multiply this with ai (since W(i) contains a i points, all with

the saIllll eigenvalues) and sum over all (EG for which W(::: Y U Wei),

1. e. over all 'with (b i = 1, ,* 1. This prove s equation (20).

To prove (21) and (22), we will need to calculate the L-class of W.

First we eval\.lste the L-clasB of the hyper surface (12). Let i denote

the inclusion of V in P ¢ and v the normal bundle, and let x ; Ht(P ~) n n

be the standard generator. Clearly v is a complex line bundle with 2 c1(v) = Ny, where y = i*x E H (Y). Therefore

To calculate the L-class of W = Y/H, we will use the theorem of §3,

We must first find the fixed-paint sets. Let ct = (aO'" ""n) E H

(here ctO=1, i.e. we have identified H with 1xH for convenience; thus

a~i 1 where bO = 1). Using the results of §6, we find

(p ¢)" Ii V n

The last equality follows because the integers 1,b1, •.• ,bn are coprime,

so for 'E: S 1 -11 J there can be at most one 1 with a i ",!:, whereas V

does not contain any point with only one non-zero coordinate. If we

renumber the coordinates so that aO"'" .:as =1, a 5+1' .•• , an" 1, we find

vI), !(zO: ... :z :O: ... :O)eP ¢ I N N

OJ, Zo + ... + Z : S n s

sO that the fixed-point set of a E: H on V is also a hypersurfaoe of

degree N. Clearly its normal bundle in V is the direct sum of n-s

copies of the restriction i*1) of the Hopf bundle 1) over P (t to Va, s

and the action of a on this normal bundle consists of multiplication

wi th a . (j:: 1, .•• , n-s) on the j th summand. We denote by x the S+J 2 A A

standard genera.tor of H (p ¢) and by y= i*x its restriction to Va(. s

(25 )

Then, in view of (23) and of the description just given of the normal

- 116 -

bundle of Va in V, we obtain from the G-signature theorem that

tanh NY ( y)S+1 n a.e 2' + Ny A ~t nh n J 2"

a y j=s+1 a j II Y -L'(a,V) (26)

It is also easy to see that j! (yr) yn-s+r, where .J is the inclusion

map from Va to V; therefore

L (a, V) j,L'(a,V) tanh Ny

Ny (27)

Summing this over a€H (recall that aO = 1) and using the theorem of §3,

we obtain

tanh N,y n ( ae2y + ) 17*1('11) I1 Y l: Ny b· ae2y _ j=O a J=1

tanh N;z: n bF

Ny n tanh b .y , (28) j",O J

where IT denotes the projection from V to V/H = Y'I, b O '" 1, and the last

line has been obtained by the usual identity.

It follows immediately that (22) holds. Indeed,

Sign W = <L(W),(W]> = TIn- <L(W),1T.[V]> = TIn- <1T*L(W),(V] >

n-1 TIrr ( coefficient of ~ in 1T*1(W) )

(this last equation holds because i*[V] € H2n- 2 (Pnc) is the Poincare

dual of Nx E li(p a:), and evaluatioh on the fundamental class in P t n n consis ts in picking out the coefYi cient of x n), and since N = I H I

n~=o b j • we obtain equation (22). But the calculation (28) can also

be used to evaluate the number Sy appearing in (21). Indeed, Y is

defined exactly like W but with one coordinate fewer, so we have

L(Y) tanh Nz Nz

n b.z

n tanh b.z j=1 J

J

where Z E li(Y) is the analogue a dimension lower of (1T*)-i y E rf(W)

(we use complex coefficients where rr* is an isomorphism). The

(29 )

complex manifold Y is embedded in W with a normal bundle whose first

Chern class is precisely E:., and G :: liN acts on this normal line bundle

by multiplication with Nth roots of unity. Finally, evaluation on [Y]

- 117 -

n-2J corresponds to finding the coefficient of y N.

G-signature theorem tells us that

2z < L (Y) (e 2 Z +1 , [Y) >

(e -1

'L N ~ tanh Nz res z::O zn-2" Nz

n

Therefore the

res z::o [dz (tanh Nz n coth b .z) (N coth Nz - coth z)] • j:01 J

This completes the proof of equation (21).

- 118 -

§18. Eguivariant signature 2! Brieskorn varieties

Lilt a (a:>., •.. , an) be an n-tuple of int egers ,,2, and

be the corresponding Brieskorn variety. The group

G (2)

th (where ~a is the group of a roots of unity) acts on Va by

Here Si or zi denotes the ith component of {, z respectively.

We will evaluate the equivariant signatures for this action by

using t he known description of the cohomology of Va and the action of G on this cohomology. We state the general result as a theorem, and single

out three special cases as corollaries. The first, obtained by taking

C = id, is Brieskorn's result for the signature of Va itself. The

seCond is a result of Hirzebruch and. Janich on the Browder-Livesay

invariant of a certain i nvol uti on on V defi ned when each a" is even. a J

The third gi ve5 the equi variant signature for the action of a certain

cyclic group embedded in G; when this action is free it embeds in a

free S1-action the signature of which will be calculated later in this

section.

To state the theorem, we use abbreviated notati on for n-tuples.

Thus if {'" (i::I.' .... ~n) e G and j = (j" •.• , j o) is an n-tuple of

integers,

o <jk < ~

"t r j f ,j1 jn we Wrl e ~ or ~1 ••. 'n. Similarly O<j<a means that

(k 1 ) d l d t ~ + • •• + .iA • = , •••• n an a eno ell 81 an

Theorem 1: The signa ture of (= (~1., .•• ,!: n) E C. on Va is gi ven by

where

[ +1, if 0< x <1 (mod 2), e:(x) = -1, if 1 < x <2 (mOd 2),

0, if x e:Z.

This can also be written as a trigonometric Bum:

Sign«(,V ) " a 1 N

- 119 -

1+t 1::

N 1-t t :::-1

n 1+ (kt-bk n

k=O 1 - I: t -bk "k

(6)

2N-1 Z

j=1 j odd

c t!.i t 1T(2~1.-j) t 1T(2s n-.il a 2N co 2a,- ",00 2an

(7)

Here N is any common multiple of a~" •• , an and bk = N/ ak , The numbers

sk in equation (7) are def'ined (mad ~) by

Corollary 1 CBrieskorn [3]); The :!ignature Of'Va i8 zero f'or .!! odd

and, for.!!- even is given by

Sign(V ) a Z O<j <a

2N-1 Z

j=1 j odd

tCa)

.J.[....i?r. j7r cot 2N cot 2a,- ••• oot2an •

(8)

(10)

Corollary 2 (Hirzebruch and Janich [19]): If 81, ••• ,ao are even, the

involution T:Va -+Va sending (z,-, .•• ,zo) to (-z,,-, .•• ,-zn) has signature

Sign(T,V) = a l:: e(l). (-1 )jl+ ••• +jn

O<j <a a

n

(-1 )2

N

2N-1 1.:

j=1 j odd

..i!!. j1T ..k cot 2N tan 2an ••• tan2an •

Corollary 3: Let N, bk be as above. The ~ acts on Va by

The signature of this action is given by

Sign{.21Tih/N ,V ) a

l:: € (..i.) O<j<a a

in+2

'" N

2N-1 ( .) '" t 1T2h- J <- co 2N

j=1 j odd

N (t =1).

...!!..i. ..!!l. cot 28.1 ••• cot 2an •

( 11 )

(12 )

( 14)

(15 )

Proof: The corollarie& are obtained from the theorem by speCializing {

- 120 -

and performing simple manipulations with the formulas. The proof that

(4) and (6) are equal is exactly like the proof given in §16 for the

special Case ,=id (i.e. for the equality of (9) and (10). Ne thus

only need to prove equation (4). To do so, we use the results on the

cohomology of Va given in Pharn [35J, Brieskorn (3 ), Hirzebruch-Mayer (20). One can give

U a

a G-e quivariant deformation retraction of Va

8 k . i Z E Va! zk ~s real and ~O (k=1, ••• , n) r .

onto

(16 )

The space Ua is naturally homeomorphic t o the join ~ai* •••• uanof the

discrete spaces ua . (j=1, ••• ,n). It follows that Va is (n-2)-connected J

and H (V) is free abelian of rank. .... , a rr(~ -1). We denote by wk the action k.

of t he generator of U on H.(Va ). ~

Let!:!. be the(n·nrimplex of Ua that

corresponds to 1EG in the identifi cation of Va with Ila: .... ~an The

group G (U ) of n-chains is Z:[ G]., where Z:( G] i B the group ring of G • .,..1' a Since the boundary operator from eli_, to Cn_7. commutes with the action

of G, the element

h = (1-w1) ... (1-wn)~ e ZZ; [G]

is a cycle. In fact h ill a generator of H (u ) as a Z:[G]-module: It'" a

~ ~-1 H,.,.~U) - Z:[G]h = Z:[GY( 1+wk+ •• ,+wk ,1~k;;n).

This describes !of (V). He can t ake a1l a basis the set of monomials "1-1 a

(18 )

wj = wit ... w~n with O<j<R, and the action of G is given in th~ obvious a -1

way by taking into account the relations 1+wk+ ••• +wk k =0. Finally,

the intersection form on H IU ) is given by 11.( a

S(xh,yh ) (x,yE .:z [GJ),

where Tj = (i-Wi)", (i-wn) E .:z[G] and E i1l defined on the generators

of Z:[G] by

::: [ 0, -0, 0,

if j,= ... =jn=O, if j,=, •• =jn=1,

otherwi!le,

(19 )

(20)

h • '" (-1 )n(n-1)/2 ( ) f were 0 For r = ri, ••• ,rn an r-tuple 0 integers

defined modulo ~, we denote by Lr the element wrh::: w~i •.• w~nh of

Hn_1 (Ua ), or rather the oorresponding element in Hn_1 (Va ) under the

- 121 -

isomorphism induced by the inclullion of' Ua in Va" Then (1 9 ) implies:

C-1)to if ! of the nuroberll sk-rk are =1 and the rest =0, S (L ,1 )

r 5 _(_1)t if ! of the numbers I5k-rk are =0 and the relit =-1,

0 otherwise.

w. define a new basis for H 1 (V ) n- a n

by

v. J

= n (1 k=1

jk + ~k wk +

(~-1)jk ~-1 ••• + ';k wk ) h

~

r mad a

21Ti/a1, where .ek '" II ',j = (jl,"" jn) is an n-tuple of integers jkfO

(21 )

(22)

(23 )

(mod ~), and ~jr denotes .e~1rl ••• ~~nrn . Then, from (21) and (23),

s r mod a

8 i (j.+k. Jr. ); .e. 1 l 1

1

if j+k '" 0 (mod a),

otherwise.

Thus the only non-zero elements of the matrix of S with respect to

the v. (where we fix ..1 by O<j<a) are the elements J

n j./2 -j./2 . /2 . /2 c. = S (v.,v ,) = 0 a1' .. a n n(.:;.l _ e. 1. ) (.;~Jl ... ~~Jn

J J -J . 1 1. 1. l= _ &j,./2 >- jn/ 2 )

'..1 •• • ~ n

e (..i.) (real, positive number). a

(24)

(25)

In particular, S is a non-degenerate form if and only if the n-tuple !

is such that ..i. = .J.. + ••• + III f ~ whenever O<j<a. a a1 an

We introduce a new basis

- 122 -

(j EM),

where M is a set of indices J with O<j<a such that, for any ~ with

O<k<a, exactly one of k, a-k is in M (this is necessary to have a

basis, since the \!Ilements (26) :5atisfy A ,= A" B ,= -B.), a-J J a-J J

We now suppose that!! is ad d.. Then the intersection form M ill;

symmetrio, and we obtain (j,k€M)

where 0jk is the Kronecker delta. Thus S is diagonal. If we write

C = (C:1.1.· •• (n) = w~1. ... w~" € G

( 2'1Tiskl a· " thus Sk = e K ; the notahon us~ng the w' 8 i8 the one intro-

(26)

(28)

duced to denote the action of G on H 1(V ), then it is clear from (23) n- a that Vj is an eig~nvalue of the action of <'; in homology:

(* v j = (29)

where <'; -j has the meaning explained before Theorem 1 and

e(j) 211 (5d1. + ... + ~). a1 an

(30 )

Therefore

r A A 0 e(J') - B, sin e(J'), "* j = j os J

If we substitute this and equations (27), (25) into the definition of

the IIquivariant signature for a manifold of dimension =0 (mod 4), we get

Sign(~,V ) a

Z 2 &(j) cos e(j). j~M

Since s(j) and co~ Q(j) are the same for ~ and a-j, we can write this

L e(j) cos e(j). O<j<a

This is equivalent to (4) since ~ is odd.

Now assume that!! is even. Then, tor j,k € M, we have

- 123 -

S (A ,,13k ) " -8 (13k , A ,) = -2i Q 'k c,' J J J J

(2d)

Therefore the matrix C:H 1 (V ) -+ H 1 (V) correllponding to S after n- a n- a

the introduction of the scalar product defined by the basis iAj' Bjlj~Ml

is (2? -20iO) with respect to this basis, where c is the diagonal ~c -

( ) ~ (-402 0 ) matrix of the c j 's jEM. Therefore CC'" = -C has the matrix 0 -4c~

wi th respect to his basis; this is posi ti ve definite since c j is pure

imaginary for even.!:! by (25). The positive definite square root of

th ' h t· (21 c I 0) Th f th t f I ' 1.8 as rna rl.X 0 21 c I' ere ore e sauare roo 0 - appear1.ng

in the definition of Sign«(,V) for a manifold V of dimension =2 (mod 4)

(see (III) of §2) has the matrix

J C/(CC*) 1/2 0 -ie/I cl ) , = ic/I cl 0 (29)

that is, J is given by

J(A) ~ o J(Bj) ~ - I I 13 o. " I I A .. OJ J OJ J (30)

,n2_1 . If we substitute (25) for C 0 and observe that 1. = -1. for even.!:!. we get

J

-e(j) 13 0' J

.IeB. ) J

c:(j) A .• J

In general, if V is a real vector space on which j:V->V is a map with

square -I, and V has a basis of the form e 1, ••• ,e r ,Je1, ••• ,Jer , then

for a map (;:V"'V commuting with J, the trace of G thought of as a

complex matrix on a oomplex vector space is

G' and G" are the matrices defined by G(e)

tr G' + i tr G", whel'e

::; 6 (G~kek + GJ'!k<Je). k J

We take for V the homology group Hn_1 (Va) and for G th'" aotion of i:. l"or e 1, ... ,e r we take iAjlj€ML The action of (; is just as in the

case of odd ll, i.e.

A, cos e(j) - B, sin e(j) J J

cas e(j) Aj + e(j)sin e(j) JA j ,

and we conclude that

Sign({;,V ) a

- 124 -

2i /, c(j) ~in 8(j). jEM

Since e(a-j)= -s(j), sin 8(a-j) = - sin Ii(j) for even,!2, this equab

i E O<j<a

<;(j) sin e(j) 1:: ( .) ilJ(j) <; J e , O<j<a

This prove s (4) for even!! and completes the proof of Theorem 1,

We now wish to study the odd-dimen~ional smooth manifold

This is diffeomorphic to the boundary of Va Ii D, where D h a disc in (!;n ak

of large radius. The advantage of using the homogeneoull equation 1: zk = 0

is that the action of ~N described in Corollary 3 of Theorem 1 now

extends to an S 1-action, defined by the /lame formula (13) with tES 1,

Recall that the G-signature theorem can be used to define an

invariant of certain group actions on odd-dimensional manifold~ (Atiyah

and Singer [1]), The definition of this "a-invariant" is as follows:

if the disjoint union of ~ copies of a G-manifold E is equivariantly

diffeomorphic to the boundary of a G-manifold X, then

CI(t,Z) i f Sign(t,X) - L'(t,X)[xtJ J

for any t€G acting i'reely on Z (so that XtlioX '" ¢). It has also

been proved (O~$a [~"J) that, if G = 81 and the action on E is fixed­

point free (Le. G is a finite subgroup of S 1 for every XEZ), then x some multiple ~ of L alwaY$ does bound an S1-manifold (we oan even

take ~ to be a power of two), and therefore the ~-invariant is defined.

It is a rational function of t which can only have poles at values of t

which have fixed-points on E.- The action of S 1 on E given by (13) is a 1

certainly fixed-point free; indeed, it is clear that tES can only have

a fixed pOint if t N:1. However, we will not be able to calculate the

a-invariant of this action on the Brieskorn manifold in general. The

problem is to find the manifold X with mEa = ax; we can take m=1 and

X = V nD all above, but the diffeomorphism of E onto V n aD can only a a a 1

be made ~N-equivariant since there is no natural action of S on Va'

There is, however, one Cai'\8 for which we can calculate th ..

a-invariant, namely when the action (13) is free. This is a general

fact: if E is a free S1-manifold, then the projection Z ~ E/S1 = Z

- 125 -

definee an S1-bundle ~ oyer Z, and we can oonsider E as the boundary

of the U :!Iooiated D2 -bundle X. Clearly X t ::: Z for t" S 1_! 1 J. Then

equation (33) gives (Atiyah and Singer [1])

cx(t,z) Sign <p te 2x + - < =..,,:---'~ L(Z), [Z] > , te2:x. _

where XElf(Z) is the first Chern class of the complex line bundle e and <p is the following quadratic form on rf-3(z) (where 2n-3=dim Z):

",(a,p) <ct i3x, [ZJ >

We now apply this with E ::: Ea (we correspondingly write Xa and Za for

the X and Z above). We first have to know when the action of the group S 1

on a is free; this is the case if the!! numbers bk ::: N/ ak are mutua.lly

coprime. The quotient Z ::: Z Is 1 is then a complex manifold. '.'re will a a

need the following facts about Za and the class

which were communicated to the author by W. D. Neumann:

Theorem 2 (Neumann, unpublished): The cohomology of Za i:!l

11(Z ) a [ ~ if i is even l

o if i is odd J

(36)

(here H'l--1 (2:a ) is free abelian of rank It j I 0 < j < a, ~ E 1L1I ; thia i& a

coneequence of the description of the intersection form (19)> and the

generator of the summand ~ in dimension 2k, denoted 4' k' is related to

the element (36) by

k x [

if 2k < n-2,

if 2k>" n-2.

Her'e d '" N/bJ. •• • b n , which is an integer since each bk divides Nand

the bk's are mutually coprime. The Chern clas~ of T(Za) equals

(39)

Using this information, we can easily evaluate the right-hand side

of (34). It follows immediately from (39) that the L-class of Za is

1(Z ) a tanh Nx

Nx

- 126 -

Furthermore, Wit have

-d,

and therefore the second term in (34) equals

2x a(t,~ ) - Sign ~ ; d· coeffioient of xn- 2 in te~ +1 L (Z )

a te,"x-1 a

[- ax te2x +1 tanh Nx n bkx ] d . re S ::-rr=T --- n

x=O x- te2x _1 Nx k=1 tanhbkx

dZJ 2z ,

2x where the last Une has been obtained by 5ubstituting z = e The

(42)

rational function in square brackets in (42) has, as well as the pole

at z=1, poles at 0, "", t- :l. , and values of ! with zN=_1. The faotors bk

z -1 in the denominator do not give new poles, since the zeroes of

these polynomials for different!'! are (except for z=1) all di:'ltinct

because the bk's are relatively prime, and the simple zero in the b

denominator at a poi nt z k = 1, z~ 1 is offset by the vanishing of

zN.1 in the numerator at such a point. All the poles except z=1 are

simple and their residues therefore easy to evaluate. Applying the

residue theorem t hen gives

n N n -bk ±.1.L 1 t - -1 t +1 1 2 + 2" - t-N +1 n ~ - N

k=1 t -1

To evaluate Sign ~, we use equation (37). We find that ~-3(Za) is

zero if n is even, while if n is odd it is isomorphic to ~ Vii th

generato~ x(n-3)/2. Since xn-2[Z ] i~ negative (eqo (41», the a

signature of the quadratiC form (35) is -1 in the latter case; thu~ (_1)n _ 1

2 Sign 'll

- 127 -

Combining (43). and (44) and using the usual identity

we obtain:

1 N z

l -1

~ zt - 1 '

Theorem 3: Let b~, ... ,bn b!l po~itive, mutually coprime integers, d a

positive integer, N = db~ ••• bn, and ~ = N/bk• The free g1-action -

on the Brieskorn manifold (32) has a-invariant (for t*1) given by

aCt, E ) a ~L

N z =-1

bk n bk _t+z l nn ..L!:....!..- - 1 + z ] ""r IT---'15k t-z k= 1 t - t k k=1 1 - z •

(46)

Notice that the values t= z in (46) do not give poles since then bk

the two produot" in the "quare braokets agree. The values t ::: 1, U 1,

also do not give poles of aCt), as one can see from the alternate

expression (43) (by the argument given above sinoe the bk'~ are coprime

and divide N). Therefore (46) defines a rational function of ! whose

only pole is at t=1, which 1.8 as it should. be for the a-invariant of

a free Circle action.

To connect this result with the signature calculations at the

beginning of this seotion, we obMrve that, for to;: I"N - !1l ,we can

caloulate the a-invariant by using the diffeomorphism from Ea to

a (Va nD), sinoe this diffeomorphism is f.JN-equivariant. Then =1 in (33)

and Sign(t,X) = Sign(t,V ) is the number caloulated in CorOllary 3 of a t

Theorem 1, 50 we only have to calculate L' (t, V )[V 1 (we can replace a a Va nD by Va everywhere if the radius of D is large enough). Because

the bk's are coprime, this is easy: the fixed-point set of t*1 is

empty unless t bk= 1 for some k and consists of a, isolated points b - b K

wi th eigenvalue s t 'j (j= 1, ••• , k, .•• , n) if t k= 1. The calculation is () ., N

the OM done in II of ~17. We find that, for t = 1, t* 1, the value

of-I.' (t, V )[vt] is precisely the value of the fir:'lt sum in (46) (i. e. of a a the sum of the first product in the square brackets), while from (15)

we see that Sign(t,V ) ill just the second sum in (46). This provide:!! a

a check on the calculations and, incidentally, an alternate proof of

the equality of (14) and (15), at least when the numbersJiare coprime. ak

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28. C. Meyer: tiber einige Anwendungen Dedekindscher SUl11lllen, J. fUr Me reine und angew. Math. 198 (1957) 143-203

29. C. Meyer: tiber die Bildung von Klasseninvarianten biruirer quadra­tischer Formen mittels Dedekindscher Summen, Abhand. Math. Sem. Hamburg 27 (1964) 206-230

30. C. Meyer: tiber die Dedekindsche Transformationsformel fUr log 1)(T), Abhand. Math. Sem. Hamburg 2Q (1967) 129-164

31. J. Milnor: Lectures on characteristic classes, notes by J. Stasheff, Princeton 1957 (mimeographed)

32. H. Osborn: Function algebras and the de Rham theorem in PL, Bull. A. M. S. 11 (1971) 386-392

33. E. Oesa: Aquivariante Cobordismustheorie, Bonn Diplomarbeit, 1957

34. E.

35. F.

Ossa: GobordisIDUstheorie von fixpunktfreien und Semifreien S1-Mannigfaltigkeiten, Bonn Dissertation, 1969

Pha.lJl: Formules de Ficard-Lefschetz generalisees at ramifications des integralea, Bull. Soc. Math. France 93 (1965) 333-367

- 130 -

36. H. Rademacher: Lectures on analytic number theory, Tate Institute of Fundamental Research, Bombay, 1954-55

37. J.-P. Serre: Groupes d'homotopie et classes de groupes abeliens, Ann. of Math. 2§ (1953) 258-294

38. E. Spanier: Algebraic topology, McGraw-Hill Book Co., New York, 1966

39. R. Thorn: Les classes characteristiques de Pontrjagin des varietes triangulees, Symp. Intern. Top. Alg., 1956, 54-67, Univ. de Mexico 1958

40. c. T. G. Wall: Surgery on compact manifolds, Academic Press, London­New York, 1970

41. D. Zagier: Higher-dimensional Dedekind sums, Math. Ann., 1972 (To appear)

42. D. Zagier: The Pontrjagin class of an orbit space, Topology, 1972 (To appear)

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