Twisted homotopy theory and the geometric equivariant 1-stem · 2016-12-14 · Twisted homotopy...

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Topology and its Applications 129 (2003) 251–271

www.elsevier.com/locate/topo

Twisted homotopy theory and the geometricequivariant 1-stem

James Cruickshank

Department of Mathematics, National University of Ireland, Galway, Ireland

Received 19 April 2001; received in revised form 6 June 2002

Abstract

We develop methods for computing the equivariant homotopy set[M,SV ]G, where M is amanifold on which the groupG acts freely, andV is a real linear representation ofG. Our approachis based on the idea that an equivariant invariant ofM should correspond to a twisted invariaof the orbit spaceM/G. We use this method to make certain explicit calculations in thedimM = dimV + dimG+ 1. 2002 Elsevier Science B.V. All rights reserved.

MSC:55P91; 55N20; 57R91

Keywords:Equivariant homotopy; Twisted homotopy; Twisted cohomology theory

1. Introduction

The following notation will be used throughout. LetG be a compact Lie group ansuppose thatM is a compact connected manifold on whichG acts freely and smoothlon the left. It is crucial here that the action is assumed to be free. LetM := M/G and letξ = (M,p, M/G) be the induced principalG-bundle. We will writeSV to denote the onepoint compactification of a realG-representation,V . SoSV = V ∪ ∞.

This paper will consider methods of analysing the set of equivariant homotopy cof equivariant maps fromM to SV , denoted[M,SV ]G. If M has nonempty boundar∂M, then we will consider[(M, ∂M); (SV ,∞)]G. We shall be particularly concerned withe differences between the equivariant problem and the corresponding nonequproblem. In [10], Peschke has shown that the classical degree classification of map

E-mail address:[email protected] (J. Cruickshank).

0166-8641/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0166-8641(02)00183-9

252 J. Cruickshank / Topology and its Applications 129 (2003) 251–271

a manifold to a sphere may be generalised to the equivariant setting, provided that onee shallfurther

ntrey

s

y withaxioms

tivation

rder tofor thence to

inld

e.m ofy asialariant

riant

s.phere

ariant

s are

keeps track of the appropriate invariants of the group actions. The methods that wdevelop cast some new light on Peschke’s results and provide a means of makingcalculations.

As a first example, suppose thatM = G× M, or that the action ofG on V is trivial.In either of these cases, it is easy to see that[M,SV ]G is isomorphic to the nonequivariahomotopy set[M,Sd ], whered is the dimension ofV . In general, these two sets adifferent. However, it is well known that[M,SV ]G is isomorphic to the set of homotopclasses of sections of a certain fibre bundle overM. If we write Γ for this set, then wemay viewΓ as a “twisted version” of the set[M,Sd ]. In Section 3 we will show how thisetΓ may be stabilised to yield a twisted version of the stable cohomotopy ofM. Thistwisted cohomotopy functor may be thought of as a generalised cohomology theortwisted coefficients and has properties corresponding to the Eilenberg–Steenrodfor generalised cohomology theories.

Sections 3, 4 and 5 contain the main results of the paper. Here, we use as mothe goal of computing[M,SV ]G when dimM = dimV + dimG + 1. We call this casethe geometric equivariant 1-stem (for reasons that will become apparent later). In odo the computation, we first develop an Atiyah–Hirzebruch type spectral sequencetwisted stable cohomotopy functor mentioned above. Applying this spectral sequethe geometric one stem, we obtain an exact sequence

Z/2→ [M,SV

]G

H1(M;L),

whereL is a certain twisted coefficient system onM. This coefficient system depends,a clearly understood way, on three things—the orientation properties of the manifoM,the action ofG onV , and the action ofG on its adjoint representation.

In Section 5, we analyse the mapZ/2 → [M,SV ]G from the above exact sequencWe show that this copy ofZ/2 corresponds, in a canonical way, to the usual one stethe sphereπk+1(S

k), k 3. In this section we also introduce twisted cobordism theora way of understanding the elements of[M,SV ]G. This method illustrates the essentunderlying geometry, and it is also in keeping with the theme that an equivariant invof M should correspond to a twisted invariant of the orbit spaceM.

Finally, in Section 6, we conclude with some explicit computations of the equivahomotopy set[(SV+n+1,A); (SV ,∞)]G, wheren is the dimension ofG. Here,V + n+ 1meansV ⊕R

n+1 andA stands for the subspace ofSV+n+1 consisting of the nonfree orbitThis set arises naturally in conjunction with the equivariant stable 1-stem of the sspectrum (see [3,10,11] for details of this).

We have also included an appendix containing some technical details on equivtriangulations that are omitted from the proof of Lemma 6.1 in the main text.

2. Preliminary remarks

One should note the following convention: Throughout this paper, all group actionleft actions (except in Section 6.1 at the very end of the paper). Thus, for example, ifA and

J. Cruickshank / Topology and its Applications 129 (2003) 251–271 253

B areG-sets, thenA×G B = A×B/∼, where(a, b)∼ (g.a, g.b) and we write[a, b] to

ction.lace anm 2.1t is, aill

y be

nce

ry

spect: Let

etesft

wistedtween

ay asktheorth

denote the equivalence class of(a, b) in this context.The interested reader should consult [7,14] for details on the material in this se

Our goal here is merely to point out that if one has nice spaces then one can repequivariant problem with corresponding twisted (or fibrewise) problem (see Theorebelow). In our context, “nice” means compactly generated in the sense of [9]. Thacompactly generated space is a weak Hausdorffk-space. Most of the spaces that we wdeal with will be finite complexes or even compact manifolds, so they will certainlcompactly generated.

Let ξ = (X,p,X) be a principalG-bundle and letF be aG-space. Recall thatξ [F ]denotes the associated fibre bundle,X ×G F →X. There is a one to one correspondebetween sections,s, of ξ [F ] and equivariant mapsφ : X → F given by the equations(p(x)) = [x, φ(x)] where x ∈ X. See [4], for example, for a proof of this elementafact.

WhenX is compactly generated, this bijection is in fact a homeomorphism with reto the compact open topologies. We can rephrase this fact in the following wayTX denote the category of spaces overX. The objects of this category are pairs(Z;f )

wheref :Z → X. A morphismg : (Z1, f1) → (Z2, f2) is a mapg such thatf2g = f1.The space of morphisms from(Z1, f1) → (Z2, f2) will be denoted mapX(Z1,Z2). It istopologised as a subspace of the map(Z1,Z2) with the compact open topology. The sof homotopy classes of morphisms will be denoted[Z1,Z2]X. We shall also have usfor the categoryT 2

X , whose objects are triples(Z,C;f ) where(Z,C) is a pair of spaceandf :Z →X. The relative mapping space mapX((Z1,C1); (Z2,C2)) is the subspace omapX(Z1,Z2) consisting of those elements that mapC1 to C2. The relative homotopy se[(Z1,C1); (Z2,C2)]X is also defined in the obvious way.

Now, (X, IdX) and(X ×G F,π) (whereπ is the projection of the fibre bundleξ [F ])are objects inTX . One observes that mapX(X, X×G F) is the same asΓ (ξ [F ]), the spaceof sections of the fibre bundleξ [F ].

Theorem 2.1. LetX be a compactly generated space. The bijection

h : mapG(X,F

)→ mapX(X,X×G F

)is a homeomorphism with respect to the compact open topologies. In particular,h inducesa bijection[

X,F]G→ [

X,X×G F]X.

We can think of the cofunctors mapX(−, X ×G F), respectively[−, X ×G F ]G, astwisted versions of the cofunctors map(−,F ), respectively[−,F ]. Thus, Theorem 2.1recasts the problem of understanding equivariant homotopy classes in terms of “thomotopy theory”. In later sections, we shall further develop this correspondence beequivariant and twisted phenomena.

Some comment is perhaps required on the use of the word twisted. The reader mwhy we do not use “fibrewise” instead? A rough guideline is that “twisted” will refer toparticular situation where the domain fibre bundle has trivial fibre. We think that it is w


making the distinction between this situation and the general fibrewise environment asample,

the

tting

e

es

pond-

tablet claim

that is

ehe

y witherfromjoints

there are certain instances where the terminology will be clarified as a result. For exthe twisted Pontryagin–Thom construction (see Section 5.1) is quite different fromfibrewise construction as presented in [1].

We remark that all of the observations in this section hold true in a relative sealso. More precisely, let(X, A) be a pair of freeG-spaces such thatA is a closedG-invariant subspace ofX, and suppose that the projectionp : X → X/G is a principalG-bundle. LetX := X/G and letA := A/G and suppose thatX is locally compact. Supposthat (F,∗) is a basedG-space and thatG fixes the basepoint. Let(X, A) ×G (F,∗) :=(X×G F, ((X×∗) ∪ (A× F))/G). Then there is a homeomorphism of mapping spac

mapG((X, A

); (F,∗)) → mapX((X,A); (X, A

)×G (F,∗)).3. Twisted stable cohomotopy

As a rule of thumb, it is generally easier to compute stable objects than the corresing unstable ones. In this section, we shall describe a stabilisation of the cofunctor[−,F ]B .Our construction will proceed at a fairly elementary level. The subject of fibrewise shomotopy theory appears to be quite a subtle one, and the author certainly does noto present the definitive version. Our goal here is merely to construct an invariantcomputable in certain cases.

3.1. Spectra over a base space

Let B be a space. A pointed object inTB is a triple(X, τ ;f ) where(X,f ) is an objectin TB andτ :B → X is a section off . A map of pointed objects is aTB -morphism thatcommutes with the sections. Given two such pointed objects(X1, τ1;f1) and(X2, τ2;f2),we define their smash product as follows:

(X1, τ1;f1)∧B (X2, τ2;f2) := (X1 ∧B X2, τ1 ∧B τ2;f1 ∧B f2).

Here,X1∧B X2 denotes the usual fibrewise smash product (see [1]), while(τ1∧B τ2)(b1)=[τ1(b), τ2(b)], and(f1 ∧B f2)[x1, x2] = f1(x1)= f2(x2).

Let εB be a trivial vector bundle of rank one overB. Let ε+B denote the fibrewise onpoint compactification of this bundle. LetS1

B denote the pointed object consisting of tbundleε+B , together with the sectionb → (b,∞) (i.e., the∞-section ofε+B ).

Definition 3.1. A prespectrumE, over B is a sequence of pointed objectsEn =(En, τn;pn), n ∈ Z, in TB , together with structure mapsσn :S1

B ∧B En → En+1.

Our prespectra overB are stable only with respect to trivial sphere bundles overB, and,for this reason, might be more accurately referred to as “naive” prespectra, by analogthe commonly accepted term “naiveG-spectrum” (see [3]). However, since in this papwe shall have use for no other kind of prespectrum, we shall omit the word “naive”our terminology. A spectrum is a prespectrum with the additional property that the ad


of the structure maps are fibrewise homeomorphisms. Of course, we have not said preciselystudyses, wehat it is

ec-

t

e

s.hism

e

ome-

atove

think

pension

what is meant by the adjoint of a fibrewise map. This would involve a more detailedof fibrewise mapping spaces and fibrewise smash products. However, for our purpocan safely ignore all these issues and just work with prespectra. We also observe tonly necessary to define a prespectrum on a cofinal subset ofZ—a slight modification isnecessary to the part of the definition concerning the structure maps.

Our main example of a prespectrum overB generalises the classical sphere (pre)sptrum, and arises in the following way. Letζ be a real vector bundle of rankd over B.We define a twisted sphere spectrum, denoted bySζ , as follows. Forn d , let Sn

ζ be

the total space of the bundle(ζ ⊕ εn−d )+ (recall that+ stands for fibrewise one poincompactification). Letpn :Sn

ζ → B be the bundle projection and letτ∞ :B → Snζ be the

∞-section of the bundle. Now(Sζ )n := (Snζ , τ∞,pn). Letσn, thenth structure map, be th

identity map (note thatS1B ∧B (ζ ⊕ εn−d

B )+B = (ζ ⊕ εn−d+1B )+B ).

3.2. Twisted generalised cohomology theories

In this section we show how prespectra overB give rise to twisted cohomology theorieLet I denote the unit interval in the real line. We fix a relative fibrewise homeomorp

(I, ∂I) × B ↔ (ε+,∞) where∞ denotes the image of the∞-section of the bundleε+. Thus, given an object(X,A;f ) in T 2

B we have fixed a canonical fibrewisrelative homeomorphism(I, ∂I) × (X,A) ↔ S1

B ∧B (X,A). Given a prespectrumE =(En, τn,pn): n ∈ Z, an object(X,A;f ) in T 2

B and an integerk, we want to define acohomology groupEk(X,A;f ). For any integerl, let

Fkl (X,A;f ) := [

(I, ∂I)l × (X,A); (El+k, im(τl+k))]

B.

It is clear that structure maps of the prespectrum, together with the fibrewise homorphism (I, ∂I) × (X,A) ↔ S1

B ∧B (X,A), induce a morphismFkl (X,A;f ) →

Fkl+1(X,A;f ). If l 2, this morphism is a homomorphism of abelian groups.

Definition 3.2. Given a prespectrumE overB, and an object(X,A;f ) in T 2B , we define

Ek(X,A;f ) := lim→· · ·→Fk

l (X,A;f )→Fkl+1(X,A;f )→·· ·.

We remark thatFkl is in fact a functor fromT 2

B to the category of abelian groups (least forl 2). Moreover, the homomorphisms from the direct limit in the definition abare in fact natural with respect to morphisms inT 2

B . Thus,Ek is also a functor fromT 2B to

the category of abelian groups.In keeping with the traditional notation for stable cohomotopy, we writeωk

ζ (X,A;f )

for Skζ (X,A;f ), whereSζ is the twisted sphere spectrum defined above. Thus, we

of ωkζ (X,A;f ) as a twisted version of the classical stable cohomotopy groupωk(X,A).

We remark that, as an immediate consequence of the fibrewise Freudenthal sustheorem (see Section 17 in [1]), we have the following:


Lemma 3.3. If (X,A) is a relative CW-complex, then the canonical stabilisation map

ed

he

e

ions

[(X,A); (Sk

ζ ,∞)]

B→ ωk

ζ (X,A;f ),

is an isomorphism ifdim(X,A) < 2k − 1 and is surjective ifdim(X,A) 2k− 1.

The following theorem shows that the functorsEk essentially behave like generaliscohomology theories with twisted coefficients.

Theorem 3.4. LetE be a prespectrum overB and suppose thatpk :Ek →B is a fibrationfor all k ∈ Z. Then, the functorsEk from T 2

B to the category of abelian groups have tfollowing properties:

(i) Homotopy invariance: If f and g are homotopic morphisms inT 2B , thenEk(f ) =

Ek(g).(ii) Exactness: There are natural transformations

δ :Ek(A;f )→Ek+1(X,A;f )

such that sequence

Ek(X;f )→ Ek(A;f )→ Ek+1(X,A;f )→ Ek+1(X;f )→Ek+1(A;f )

is exact for allk.(iii) Excision: Let A1 and A2 be subspaces ofX, with interiors A′

1 respectivelyA′2.

Suppose thatX =A′1 ∪A′

2. Then the excision map,

Ek(X,A2;f )→Ek(A1,A1 ∩A2;f |A1),

is an isomorphism.(iv) Additivity: Let (X,A;f ) be an object such thatX = ⋃

λ Xλ where each subspacXλ containsA and the subspacesXλ − A are pairwise disjoint. Letfλ denotethe restriction off to Xλ. Then the homomorphisms induced by the inclus(Xλ,A;fλ) → (X,A;f ) representEk(X,A;f ) as a direct product. That is,

Ek(X,A;f )∼=∏λ

Ek(Xλ,A;fλ).

(v) Local triviality: Ek(X,A;−) is a functor from the fundamental groupoid,Π1(map(X,

B)), to the category of abelian groups. Moreover, suppose thatH : I ×X → B is ahomotopy withH0 = f andH1 = g, then the diagram

Ek(A;f )

δ

(H |A)∗Ek(A;g)

δ

Ek+1(X,A;f )H∗ Ek+1(X,A;g)

commutes.


Proof. As before, letFk(X,A;f ) := [(I, ∂I)l×(X,A); (El+k, im(τl+k))]B . The proof of

or

d

ok at

or

isr. Thus

.

d

iv) areeralised.4,cal

s

bruchstable

l

the theorem consists of checking various properties of the unstable functorsFkl , and then

obtaining the required properties ofEk by passing to the direct limit. Clearly, the functFk

l is invariant with respect to homotopies of morphisms inT 2B , so property (i) follows

by taking direct limit overl. Properties (iii) and (iv) follow in a similarly straightforwarfashion and we can safely leave the details to the reader.

We shall concentrate on the remaining two items to be checked. First, we loexactness (property (ii)). In order to construct the connecting homomorphismδ, we mustconstruct a morphismδl :Fk

l (A;f )→ Fkl (X,A;f ) and then pass to the direct limit. F

the sake of clarity, we shall consider the casel = 0. The details for arbitraryl are essentiallythe same. So, letc :A→Ek represent an element ofFk

0(A;f ). Let c denote the followingcomposite:

(I, ∂I)×AS1B∧Bc

S1B ∧B Ek

σk Ek+1.

Certainly, c sends∂I × A to im(τk+1). That is to say, ift = 0 or 1, thenc(t, a) =τk+1(f (a)). Thus, we can extendc to a fibrewise mapc : 0 × X ∪ I × A → Ek+1by setting c(0, x) := τk+1(f (x)). Clearly, c sends 0× X ∪ ∂I × A to im(τk+1). Now,pk+1 :Ek+1 →B is a fibration, so by the fibrewise homotopy extension property,c extendsto a fibrewise mapc : I ×X→Ek+1. The restriction ofc to 1×X sends 1×A to im(τk+1),so c|1×X represents an element ofFk+1

0 (X,A;f ). One must check that this elementindependent of the choices made in its construction and we leave this to the readewe have a well defined morphismδ0 :Fk

0(A;f ) → Fk+10 (X,A;f ). The morphismsδl ,

l 1 are constructed in a similar fashion, andδ is obtained by passing to the direct limitIt remains to prove the local triviality property (property (v)). LetH be a homotopy

as in the statement of the theorem. The homotopy lifting property of the mappk :Ek →B ensures thatH induces a morphismH∗,l :Fk

l (X,A;f ) → Fkl (X,A;g). Passing

to the limit yields a morphismH∗ :Ek(X,A;f ) → Ek(X,A;g). Once again, it isstraightforward (but tedious) to check thatH∗ has all the required functoriality annaturality properties.

Some remarks on Theorem 3.4 are perhaps in order. Of course, properties (i)–(essentially exact analogues of the Eilenberg–Steenrod axioms that characterise gencohomology theories. Thus, if the spectrumE satisfies the conditions of Theorem 3we shall refer to the functorE∗ as a twisted generalised cohomology theory. The lotriviality property is so named because of the following observation: Iff is homotopic toa constant mapcb0, that sends everything to a fixed pointb0 ∈B, then property (v) ensurethat Ek(X,A;f ) is isomorphicEk(X,A; cb0). Now the functorE∗(−,−; cb0) is just ageneralised cohomology theory in the classical sense.

3.3. The Atiyah–Hirzebruch spectral sequence

In this section we will see that twisted cohomology theories have an Atiyah–Hirzetype spectral sequence. Later, we will apply this spectral sequence to the twistedcohomotopy functor,ω∗

ζ , that we introduced in Section 3.2.


LetE∗ be a twisted generalised cohomology theory as in Theorem 3.4, and let(X,A;f )

tm

n

lassicalt

ce

veryplex

a

m of

ncepectralsafely

rgence

be an object inT 2B . The mapf :X → B gives rise to a family of local coefficien

systemsLq , q ∈ Z, on X, in the following way: Recall that a local coefficient systeon X is a functor from the fundamental groupoid ofX, to the category of abeliagroups. Thus, givenx ∈ X, we defineLq(x) = Eq (∗; cf (x)), wherecy :∗ → y ∈ B. Byproperty (v) of Theorem 3.4,Lq does indeed define a functor fromΠ1(X) to abeliangroups. Recall that in the remarks after Theorem 3.4, we observed that there is a cgeneralised cohomology theorye corresponding to the twisted theoryE∗. We remark thathe underlying abelian group of the coefficient systemLq is justeq(∗).

Theorem 3.5. Let (X,A;f ) be an object inT 2B and suppose that(X,A) is a relative

CW-complex. The skeletal filtration of(X,A) induces a cohomological spectral sequenE(X,A;F) whoseE2 page is given by

Ep,q2 (X,A;f )∼=Hp

(X,A;Lq

),

whereH stands for ordinary cellular cohomology(with local coefficients). Moreover, if(X,A) is finite-dimensional, then this spectral sequence converges toE∗(X,A;f ).

Proof. The proof follows the pattern of the classical untwisted version (see [6])closely so we shall omit many of the routine details. Note that the cellular cochain comof (X,A) with coefficients inLq is

· · ·→Hk(Xk,Xk−1;Lq

) →Hk+1(Xk+1,Xk;Lq) →·· · .

The differential is the composition of the canonical mapHk(Xk,Xk−1;Lq)→Hk(Xk;Lq)

with the connecting homomorphism associated to the pair of CW complexes(Xk+1,Xk)

(see Chapter VI of [13] for details). The groupHk(Xk,Xk−1;Lq) can be represented asdirect product∏

λ∈Λk

Hk(Dk,Sk−1;h∗λLq

),

whereΛ is an indexing set for the set ofk-cells ofX andhλ : (Dk,Sk−1)→ (Xk,Xk−1) isthe characteristic map for theλth k-cell.

Now, sinceDk is contractible to 0 we can, in a canonical way, find an isomorphish∗λLq to the coefficient system which is constant ath∗λLq (0) = Eq(∗; cf (hλ(0))). TheE1page of the spectral sequence associated to the skeletal filtration of(X,A) is given by

Ep,q1 =Ep+q

(Xp,Xp−1;f )

.

But Ep+q(Xp,Xp−1;f ) ∼= ∏λ∈Λp

Ep+q(Dp,Sp−1;f hλ) by property (iv) of Theo-rem 3.4, and using property (v) of Theorem 3.4, we have

Ep+q(Dp,Sp−1;f hλ

)∼=Ep+q(Dp,Sp−1; cf (hλ(0))

) ∼=Eq(∗; cf (hλ(0)))

wherecz denotes the constant map atz. The right hand isomorphism in the above sequeis the suspension isomorphism. Thus we need only check that the differential of the ssequence corresponds to the differential of the cellular chain complex. This may beleft to the reader, as it follows closely the pattern of the untwisted case. The conve


of the spectral sequence when(X,A) is finite-dimensional follows from the fact that the

ed

,

e

that

ction

ve a

to

t onerphismnce in

ur in

hat

associated filtration ofE∗(X,A;f ) is finite in this case.

4. The twisted stable 1-stem

We shall consider the following problem: LetM be a smooth compact connectmanifold on whichG acts freely. LetV be a real linear representation of dimensiond . Whatcan we say about[(M, ∂M); (SV ,∞)]G? Recall thatξ denotes the principalG-bundleM → M := M/G, and thatSd

ξ [V ] is the total space ofξ [V ]+. As observed in Section 2

we can consider the twisted cohomotopy set[(M,∂M); (Sdξ [V ],∞)]M instead of the abov

equivariant homotopy set. Also, as a consequence of Lemma 3.3, ifm := dimM < 2d −1,then [

(M,∂M); (Sdξ [V ],∞

)]M

∼= ωdξ [V ](M,∂M; IdM).

The prespectrumSξ [V ] satisfies the hypothesis of Theorem 3.4, so all the machinerywe developed in Sections 3.2 and 3.3 applies to the functorωξ [V ]. In this section, we willuse the Atiyah–Hirzebruch spectral sequence to analyse the groupωd

ξ [V ](M,∂M; IdM) inthe case whered =m− 1. This case is the twisted stable 1-stem referred to in the setitle.

Certainly, (M,∂M) has a finite CW decomposition, so by Theorem 3.5, we haspectral sequenceE, converging toωd

ξ [V ](M,∂M; IdM), whoseE2 page is given by

Ep,q

2∼=Hp

(M,∂M;Lq

).

Here,Lq is a coefficient system onM whose underlying abelian group is isomorphicωq(S0) (i.e., theq th stable cohomotopy group of the sphereS0). Let m= dimM. Clearly,this spectral sequence vanishes forp > m and forq > 0 (sinceωq(S0) is trivial if q > 0).As particular consequences of these observations, we have the following:

Lemma 4.1. There is an isomorphism

ωmξ [V ](M,∂M; IdM)∼=Hm

(M,∂M;L0).

Lemma 4.2. There is an exact sequence

Hm(M,∂M;L−1) → ωm−1

ξ [V ] (M,∂M; IdM)→Hm−1(M,∂M;L0) → 0.

We shall not have further use for Lemma 4.1. However, it is worth remarking thacan recover some of the results in [10], on equivariant degree theory, from the isomoin Lemma 4.1. We shall concentrate on the consequences of the exact sequeLemma 4.2. Our goal is to identify the two ordinary cohomology groups that occthe exact sequence.

First, consider the groupHm(M,∂M;L−1). The coefficient systemL−1 has un-derlying abelian groupZ/2. SinceM is connected, we can immediately deduce tHm(M,∂M;L−1) ∼= Z/2. The analysis ofHm−1(M,∂M;L0) is slightly more involved.Let Θ be the orientation bundle ofM. We may viewΘ as a coefficient system onM (see


Chapter VI of [13]). So, forx ∈M, Θ(x)=Hm(M,M− x;Z). Poincaré duality says that,

uality.

hismingOur

ext for

l

ss

ll

f

of

rf

for any coefficient systemL onM, there is an isomorphism

Hm−p(M,∂M;L)∼=Hp(M;L⊗Θ).

The interested reader can consult [12] for a thorough account of Poincaré dCombining these observations with Lemma 4.2, we obtain:

Lemma 4.3. There is an exact sequence

Z/2→ ωm−1ξ [V ] (M,∂M; IdM)→H1

(M;L0 ⊗Θ

)→ 0.

5. Twisted framed cobordism

Our motivation in this section is to gain a better understanding of the homomorpZ/2 → ωm−1

ξ [V ] (M,∂M; IdM) from Lemma 4.3. We shall use this objective as a jumpoff point to introduce another “twisted” invariant, namely twisted framed cobordism.basic idea is to gain some geometric understanding of the elements ofωm−1

ξ [V ] (M,∂M; IdM)

in the same way that classical framed cobordism theory provides a geometric cont(untwisted) cohomotopy theory.

As above, suppose thatM is a smooth compact connected manifold of dimensionm,with boundary∂M. If N is a submanifold ofM, then νM(N) will denote the normabundle ofN in M. Suppose thatζ is a smooth vector bundle of rankd over M. A ζ -framed submanifold ofM is a pair(N,φ), whereN is a smoothly imbedded boundarylesubmanifold ofM andφ is a smooth bundle isomorphism fromνM(N) to ζ |N . Note thatthe dimension ofN is necessarilym−d . Let pr :I×M →M be the projection. We can puζ back along this projection map to obtain the bundle pr∗ ζ over I ×M. Given(N1, φ1)

and (N2, φ2), ζ -framed submanifolds ofM, we say that they areζ -framed cobordant ithe following is true: There exists a smoothly imbedded submanifoldW ⊂ I × M anda smooth bundle isomorphismΦ : νI×M(W) → pr∗ ζ |W such that,∂W =W ∩ (∂I ×M),W ∩(0×M)= 0×N1 andW ∩(1×M)= 1×N2. Moreover, we require thatΦ|0×N1 = φ1andΦ|1×N2 = φ2. One can check thatζ -framed cobordism is an equivalence relation.

Definition 5.1. We defineΩdζ (M,∂M) to be the set ofζ -framed cobordism classes

ζ -framed submanifolds ofM.

Clearly, whenζ is a trivial bundle of rankd , Ωdζ (M,∂M) is isomorphic toΩd(M,∂M),

the set of framed cobordism classes of(m − d)-dimensional submanifolds ofM. Itis a well-known fact thatΩd(M,∂M) ∼= [(M,∂M); (Sd,∞)], so we expect a similaisomorphism in the twisted setting. Recall (see Section 3.1) thatSd

ζ is the total space otheζ+ (the fibrewise one point compactification ofζ ) and that we write∞ for the imageof the∞-section,τ∞, of ζ+.

Lemma 5.2. Let ζ be a smooth vector bundle of rankd overM. Then

Ωdζ (M,∂M)∼= [

(M,∂M); (Sdζ ,∞

)]M.


Proof. The proof is completely analogous to that of the untwisted version mentioned

ow. Itimply

cesct

ee,s

f

above. Given an element[f ] ∈ [(M,∂M); (Sdζ ,∞)]M one may assume thatf is smooth

and transverse to the 0-section ofζ+. ThusN := f−1(0) is a submanifold ofM andthe derivative off induces aζ -framing of N . Conversely, given(N,φ), a ζ -framedsubmanifold ofM, one constructs a mapf ∈ mapM((M,∂M); (Sd

ζ ,∞)) using a twistedversion of the Pontryagin–Thom construction that we describe in Section 5.1 belremains to check that these two constructions provide the required bijection. This is sa matter of modifying the untwisted argument appropriately.

We can use this lemma to identify the image of the homomorphismZ/2 →ωm−1ξ [V ] (M,∂M; IdM) from Lemma 4.3. LetU be a (closed) disc neighbourhood inM, with

interiorU . The inclusioni : (M,∂M)→ (M,M −U) induces a map

i∗ :ωm−1ξ [V ] (M,M −U ; IdM)→ ωm−1

ξ [V ] (M,∂M; IdM).

Now ωm−1ξ [V ] (M,M − U ; IdM) ∼= ωm−1

ξ [V ] (U, ∂U ;f ), wheref :U → M is the inclusion.

Moreover, sinceU is contractible,f is homotopic to a constant mapc. Thus,

ωm−1ξ [V ] (M,M −U ;f )∼= ωm−1

ξ [V ](U,∂U ; c)∼= ωm−1(U,∂U

)∼= Z/2.

Lemma 5.3. The following diagram commutes:

ωm−1ξ [V ] (M,M −U ;f ) i∗

∼=

ωm−1ξ [V ] (M,∂M; IdM)

Z/2

j

wherej is the homomorphism from Lemma4.3.

Proof. We can giveM a CW decomposition so thatM − U is a subcomplex ofM.Then the inclusion(M,∂M) ⊂ (M,M − U) induces a morphism of spectral sequenE(M,M − U ;f ) → E(M,∂M; IdM) and this in turn induces a morphism of exasequences

Hm(M,∂M;L−1)j

ωm−1ξ [V ] (M,∂M; IdM) Hm−1(M,∂M;L0)

Hm(M,M −U ;L−1)k

∼=

ωm−1ξ [V ] (M,M −U ;f )

i∗

Hm−1(M,M −U ;L0)

Now k is the compositionEm−1,02 → E

m−1,0∞ → ωm−1ξ [V ] (M,M − U ;f ) associated to th

spectral sequenceE(M,M −U ;f ). If we examine theE2 page of this spectral sequencwe see thatEm−1,0

2 = Em−2,02 = 0. Thus k is an isomorphism. The lemma follow

immediately from this. Now, as we have already remarked, the groupωm−1

ξ [V ] (M,M−U ;f ) is in fact isomorphic

to ωm−1(U, ∂U) ∼= πm(Sm−1) if m 4. Thus, assuming thatm 4, any construction o


the nontrivial element ofπm(Sm−1) will yield a construction of the image underj of the

of

e

lelint3.3

g

is in

to inctionamednd are

apodle

er

tical

nontrivial element ofZ/2. For example, letN0 be a 1-dimensional submanifold ofU that isdiffeomorphic toS1. Now U is contractible, soξ [V ]|U is isomorphic to the trivial bundleεm−1 :U × R

m−1 → U . Thus,ξ [V ]-framings of the normal bundle ofN0 correspond toordinary framings of the normal bundle ofN0 and

Ωm−1ξ [V ]

(U,∂U

) ∼=Ωm−1εm−1

(U,∂U

) ∼= [(U,∂U

); (Sm−1,∞)]∼= Z/2.

So, letφ0 be theξ [V ]-framing ofN0 such that(N0, φ0) represents the nontrivial elementΩm−1

ξ [V ] (U, ∂U). Clearly,(N0, φ0) represents an element ofΩm−1ξ [V ] (M,∂M). We can apply

the Pontryagin–Thom construction (described in Section 5.1 below) to(N0, φ0) to obtainf0 ∈ mapM((M,∂M); (Sm−1

ξ [V ] ,∞)). Then[f0] ∈ ωm−1ξ [V ] (M,∂M; IdM) represents the imag

of the nontrivial element ofZ/2, under the homomorphismj .It is worth pointing out that the element[f0] can be trivial. In fact, this is even possib

in the untwisted case as the following example demonstrates: Letζ be a nontrivial reavector bundle of rank 3 overS2 and letM be the total space of the fibrewise one pocompactificationζ+. M is a simply connected 5-dimensional manifold. By Lemmaand Lemma 4.3, we see that[M,S4] ∼= ω4(M) ∼= 0 or Z/2. We can identifyS2 with the0-section of the bundleζ+. Let N be the equator ofS2. Now N is the boundary ofS2+,the northern hemisphere ofS2 and this fact gives rise to a framing ofνM(N). However,N is also the boundary of the southern hemisphere,S2−. This gives rise to another framinof νM(N) and the nontriviality of the bundleζ implies that these two framings ofνM(N)

must homotopically distinct. A little further reflection will convince the reader that thfact means that[M,S4] ∼= 0.

5.1. The Pontryagin–Thom construction

We will describe the twisted Pontryagin–Thom construction that was referredthe proof of Lemma 5.2. We remark that this is different from the fibrewise construdescribed, for example, in [1]. The fibrewise version is concerned with fibrewise frsubmanifolds of a fibrewise manifold over some base space. We, on the other haconsidering the situation where the base space is the manifold and we have aζ -framing ofsome submanifold.

Let (N,φ) be aζ -framed submanifold ofM. We must construct a corresponding mf :M → Sd

ζ that is a section of the bundleζ+. Suppose that we fix a tubular neighbourhoU of the submanifoldN . Thus, we can identifyU with the total space of the normal bundνM(N). Let q :U → N be the bundle projection of this normal bundle. LetE denote thetotal space of the bundleζ and letp be the bundle projection. LetE|N , respectivelyE|U ,be the total spaces of the bundlesζ |N , respectivelyζ |U . We have a homeomorphism (ovN ) φ :U → E|N and we can use this to identifyE|U with E|N ⊕N E|N as follows: Thebundle projectionq :U →N is a homotopy equivalence, so there is a unique (up to verhomotopy) bundle map

E|U h E|N

U N


such thath is a linear isomorphism when restricted to fibres overU . Now, Consider the

y. The

s

en thesmXV of

inf

sy.

ncef

commutative solid diagram

E|Uh

γp

U

φ

E|N ⊕N E|N β

α

E|Np

E|N p N

whereα andβ are the projections onto the first and second summands respectivelbottom right square of this diagram is a pullback square, so we obtain a mapγ :E|U →E|N ⊕N E|N . Clearly,γ is a bundle morphism that covers the homeomorphismφ :U →E|N , so in order to check thatγ is a homeomorphism, we need only check that kerγ = 0.This follows immediately from the fact thath is a linear isomorphism on fibres overU .Now, one can construct the required mapf :M → Sd

ζ in the following way: Ifm /∈U , then

letf (m) := τ∞(m) and ifm ∈ U , letf (m) := γ−1∆φ(m), where∆ :E|N →E|N ⊕N E|Nis the diagonal map (i.e.,∆(v)= (v, v)). The following calculation shows thatpf = IdM .Form ∈U

pf (m)= pγ−1∆φ(m)= φ−1α∆φ(m)= φ−1IdE|N φ(m)=m.

Also, it is clear from the construction thatf |U is a proper map, sof is continuous. Onecan check that the fibrewise homotopy class of the mapf is independent of the choicemade in the construction.

5.2. Equivariant framed cobordism

We conclude this section with some remarks concerning the relationship betwetwisted framed cobordism ofM from Definition 5.1 and the equivariant framed cobordiof M . For more details on equivariant cobordism, the reader is referred to Chapter[8].

If V is a representation ofG andX is anyG-space, then letεVX denote theG-equivariantvector bundleX × V → X. A V -framed submanifold ofM is a pair(N, φ), whereN isa smoothly imbedded boundarylessG-invariant submanifold ofM andφ : νM(N) → εVNis an equivariant bundle isomorphism. We defineV -framed cobordisms of such pairsthe obvious way (compare with Definition 5.1) and LetΩV

G(M, ∂M) denote the set oV -framed cobordism classes ofV -framed submanifolds ofM. Our goal in this section toshow that theξ [V ]-framed cobordism ofM is the twisted invariant ofM that correspondtheV -framed cobordism ofM . That is the content of the following lemma and corollar

Lemma 5.4. Let ζ = (E, p, M) be an equivariant vector bundle overM and let ζ =(E/G, p/G,M/G) =: (E,p,M). Then there is a canonical one to one correspondebetween the set of equivariant vector bundle isomorphisms fromεV

Mto ζ , and the set o

vector bundle isomorphisms from the bundleξ [V ] to ζ , whereξ is the principal bundleM →M.


Proof. It is quite clear that given an equivariant bundle isomorphismh : εV → ζ , thatenr

f

age ofitingena in

agin–

m

on of

s ofion, iff

l

n

Mthis will factor to a bundle isomorphismh : ξ [V ] → ζ . Conversely, suppose we are givh : ξ [V ]→ ζ we can construct the correspondingh using the pullback property of the loweright hand square of the following commutative diagram.

M × V

h

M ×G V

h

E E

M M

It is a simple matter to verify thath must be equivariant. Corollary 5.5. Let M , M, ξ andV be as above. Then

ΩVG

(M, ∂M

) ∼=Ωdξ [V ](M,∂M).

Proof. The lemma guarantees a correspondence betweenV -framings of submanifolds oM , andξ [V ]-framings of submanifolds ofM.

Of course Corollary 5.5 is just a special case of Theorem 2.1, restated in the langucobordism. But in the spirit of the rest of the paper, we felt that it was worthwhile exhibexplicitly, once again, the correspondence between twisted and equivariant phenomthis context.

Also, Corollary 5.5 provides us with another way to understand the twisted PontryThom construction. If(N,φ) is a ξ [V ]-framed submanifold ofM, let (N, φ) be thecorrespondingV -framed submanifold ofM. Now, applying the classical Pontryagin–Thoconstruction to(N, φ) yields an equivariant mapf : (M, ∂M) → (SV ,∞). Clearly, finduces a fibrewise mapf : (M,∂M) → M ×G (SV ,∞). This mapf is the same (upto homotopy) as the map resulting from the twisted Pontryagin–Thom constructiSection 5.1.

6. Some equivariant computations

In this section, we will conclude the paper with some explicit computationequivariant homotopy sets (as promised in the introduction). Throughout this sectW is any representation ofG, then we will writeA for the space of nonfree orbits oSW . Also, throughout this section we will writek to indicate the trivialk-dimensional rearepresentation ofG, wherek is some nonnegative integer. Thus, for example,V + n+ 1stands for the representationV ⊕R

n+1 with G acting trivially on the second summand.Let V be a representation ofG of dimensiond . We will apply the theory developed i

the preceding sections to the equivariant homotopy set[(SV+n+1,A

); (SV ,∞)]G,


where n = dimG. As we mentioned in the introduction, there is a natural map

freens of5].

ts

-

thatnely,

m

ngthynuity,

ale

sef twonamely

[(SV+n+1,A); (SV ,∞)]G → ωG1 (S0) that is a monomorphism for “large”V . For details

of this, see [3,10,11].The following technical lemma and corollary reduce the problem to one involving a

G-manifold. The proof of the lemma relies on the theory of equivariant triangulatioG-manifolds. For the appropriate background on this topic the reader is referred to [

Lemma 6.1. If W is a real representation ofG with at least one free orbit, then there exisU , aG-invariant open neighbourhood ofA in SW , with the following properties.

(1) The inclusion of pairs(SW ,A) → (SW ,U) is an G-equivariant homotopy equivalence.

(2) M := SW −U is a smoothG-submanifold ofSW with boundary∂M = M ∩U .

Here,U denotes the closure ofU in SW .

Proof. From [5], we know that the compactG-manifold SW can be equivariantlytriangulated. Thus, we can find a homeomorphismh :L → SW/G whereL is a finitesimplicial complex andh has the equivariant lifting property. In particular, this meansthere is a subcomplexK ⊂ L such thath(K)=A/G. Intuitively, we use this triangulatioto find a “regular” neighbourhood ofA that has the required properties. More precislet t be ann-simplex in L and suppose thats is anm-simplex in t ∩ K (m n). Letn := (x0, . . . , xn) ∈ R

n+1: xi 0,∑

xi = 1 be the standardn-simplex. We can identifym with the space(x0, . . . , xm,0, . . . ,0) ∈ n, and we can fix a linear isomorphisα :n → t so thatα(m)= s. Let

Om :=(x0, . . . , xn) ∈n: x0 + x1 + · · · + xm >

1

2

and let Os,t := α(Om). If s = t , let Os,t := s. Now, observe thatOs,t is an openneighbourhood ofs in t . For each simplexs in L, let Os := ⋃

t⊃s Os,t where the unionis taken over all simplices ofL that contains and letO := ⋃

s∈LOs . Finally, let Ube the preimage ofO under the quotient mapSW → SW/G. We must check thatUhas the required properties. This involves some straightforward but relatively lemanipulations with triangulations and simplices, so in the hope of maintaining contiwe relegate the details to Appendix A.

The particular case of Lemma 6.1 whereG= SO2 andW is the canonical 2-dimensionreal representation ofSO2 is illustrated in Fig. 1. In this case,SW is a copy of the 2-spherandA is the set consisting of the north and south poles (labelledn and s respectively).The shaded areas (the polar ice caps) representU . One might also observe that in this cathe manifoldSW has an equivariant triangulation (in the sense of Illman) consisting o1-simplices, namely the northern and southern hemispheres, and three 0-simplices,the north and south poles and the equator.

Corollary 6.2. LetW andM be as in Lemma6.1and let(F,∗) be a basedG-space. Then[(SW ,A); (F,∗)]G ∼= [(M, ∂M); (F,∗)]G.


a

e

ain

ly

.

Fig. 1. Lemma 6.1 whenG= SO2 andW is the canonical 2-dimensional representation ofSO2. The shaded arerepresentsU .

Proof. The inclusion(SW ,A) → (SW ,U) is a G-equivariant homotopy equivalencand thus induces a bijection[(SW ,U); (F,∗)]G → [(SW ,A); (F,∗)]G. But it is alsoclear that the inclusion(M, ∂M) → (SW ,U) induces a bijection[(SW ,U); (F,∗)]G →[(M, ∂M); (F,∗)]G.

Now, we will apply Corollary 6.2 in the caseW = V + n+ 1 wheren is the dimensionof the groupG andV is a realG-representation of dimensiond . Let M be the manifoldcontained withinV + n + 1 whose existence is asserted by Lemma 6.1 and letξ be theprincipal bundleM →M =: M/G. Combining Corollary 6.2 and Theorem 2.1 we obta bijection[(

SV+n+1,A); (SV ,∞)]

G∼= [

(M,∂M); (Sdξ [V ],∞

)]M.

Now, letm := dimM and observe thatm= d+1. If m 4, then by Lemma 3.3 (essentialby the fibrewise Freudenthal suspension theorem),[

(M,∂M); (Sm−1ξ [V ] ,∞

)]M

∼= ωm−1ξ [V ] (M,∂M; IdM).

Now, combining these observations with Lemma 4.3, we obtain an exact sequence

Z/2→ [(SV+n+1,A

); (SV ,∞)]G

H1(M;L0 ⊗Θ

), (1)

whereL0 is the coefficient system onM induced by the map IdM (as described inSection 3.3) andΘ is the orientation bundle of the manifoldM.

Now, we fix a basepointm0 of M. Note that ifJ is any coefficient system onM, thenJ is determined by a homomorphismπ1(M,m0)→ Aut(J (m0)) (sinceM is connected)We will denote the kernel of this homomorphism byKJ , and its abelianisation byKab

J .

We shall use the following lemma to compute the homology groupH1(M;L0 ⊗Θ).

Lemma 6.3. If J (m0)∼= Z, then there is an isomorphism

H1(M;J )∼=KabJ /R,

whereR is the subgroup ofKabJ generated by the seta2: a ∈ π1(M)−KJ .

We will defer the proof of this result to the end of the section.


LetG0 be the connected component of the identity element ofG. Let Ad(G) denote the

te

t

th

f

.act

adjoint representation ofG. We will now see that, ifM is simply connected, thenKL0⊗Θ

can be identified with a certain subgroup ofG/G0.

Lemma 6.4. If the space of free orbits ofV is simply connected, thenπ1(M) ∼= G/G0.Moreover, under this identification,KΘ is the subgroup ofG/G0 that consists of theelements which either preserve orientation on bothV andAd(G), or reverse orientationon bothV andAd(G).

Proof. The isomorphismπ1(M,m0) ∼= G/G0 follows immediately from the long exacsequence of homotopy groups induced by the fibrationG→ M →M. We observe that thisomorphism can be described as follows: Given a pathα : (I, ∂I)→ (M,m0), we can liftα to a pathα : I → M such thatα(1)= g.α(0) for someg ∈G. ThengG0 is the elemenof G/G0 corresponding to[α] ∈ π1(M.m0). Now, it is shown in [10] that[α] ∈KΘ if andonly if g either preserves orientation on bothM and Ad(G), or reverses orientation on boM and Ad(G). Lemma 6.5. With the same hypothesis and notation as Lemma6.4, KL0 is the subgroup oG/G0 that acts by orientation preserving maps onV .

Proof. Let α : (I, ∂I) → (M,m0) and letgG0 be the corresponding element ofG/G0(as above). Recall thatL0(m) := ω0

ξ [V ](∗; cm), wherec :∗ → m. So it is clear thatα

acts trivially onL0(m0) precisely whenα preserves orientation on the bundleξ [V ]. Thishappens if and only ifg preserves orientation onV . Lemma 6.6. With the same hypothesis and notation as Lemma6.4, KL0⊗Θ is the subgroupof G/G0 consisting of elements which preserve orientation onAd(G).

Proof. The groupKL0⊗Θ consists of those elements ofG/G0(= π1(M)) that preserveorientation on(L0⊗Θ)(m0)= L0(m0)⊗Θ(m0). ThusgG0 ∈KL0⊗Θ if eitherg preservesorientation on both ofL0(m0) andΘ(m0), org reverses orientation on both ofL0(m0) andΘ(m0). Thus,

KL0⊗Θ = (KL0 ∩KΘ) ∪ ((G/G0 −KL0)∩ (G/G0 −KΘ)

).

The result now follows immediately from Lemmas 6.4 and 6.5.Theorem 6.7. LetG be a compact Lie group of dimensionn. LetV be aG representationSuppose thatdimV 3 and thatSV − A is simply connected. Then there is an exsequence

Z/2→ [(SV+n+1,A

); (SV ,∞)]G

H,

whereH is the subquotient ofG/G0 defined as follows: LetK be the subgroup ofG/G0that preserves orientation onAd(G). Then

H :=Kab/⟨a2: a ∈G/G0 −K

⟩.


Proof. This follows immediately from Lemmas 6.6, 6.3 and the exact sequence (1).

rouproup

lent to

up

ctt

-

pose

sm

mt

a

ef

6.1. Proof of Lemma 6.3

We remark that since Lemma 6.3 is only concerned with the first homology gof a connected manifoldM, it may be rephrased as a result concerning the gtheoretic homology of the fundamental group ofM. Thus, in the following,H1 standsfor group theoretic homology. One can easily see that Corollary 6.9 below is equivaLemma 6.3.

Lemma 6.8. Let W be a group and letI be the augmentation ideal of the integral groring Z[W ]. Suppose thatL is a (left) W -module. ThenH1(W ;L) is isomorphic to thekernel of the multiplication mapI ⊗W L→ L.

Proof. Let Z denote the integers with trivial rightW action. There is a short exasequence of rightW -modulesI Z[W ] Z. Tensoring this withL yields an exacsequence

TorW(Z[W ],L) → TorW(Z,L)→ I ⊗W L→ L Z ⊗W L.

Now, TorW(Z[W ],L) = 0 andH1(W ;L)∼= TorW(Z,L), so Lemma 6.8 follows immediately. Corollary 6.9. Suppose that the underlying abelian group ofL is isomorphic toZ and letK be the subgroup ofW that acts trivially onL. Then

H1(W ;L)∼=Kab/⟨a2: a ∈W −K

⟩.

Proof. If K = W , then this is immediate from the standard Hurewicz theorem. Supthat [W : K] = 2. Let e denote the identity element ofW and let 1 be a generator ofL.Observe thatI ⊗Z L is a free abelian group with basis(e−w)⊗1: w ∈W, w = e. NowI ⊗W L is a quotient ofI ⊗Z L. Moreover, the kernel of the quotient homomorphiis generated by the set(e − wk) ⊗ 1 − (e − w) ⊗ 1 − (e − k) ⊗ 1: w ∈ W, k ∈K ∪ (e−wy)⊗ 1− (e− y)⊗ 1+ (e−w)⊗ 1: w ∈W, y ∈W −K. Thus, ifF is thefree abelian group on the set of symbolsw: w ∈W , there is a surjective homomorphisρ :F → I ⊗W L that sendsw → (e − w) ⊗ 1. The kernel ofρ is generated by the seR := wk −w− k: w ∈W, k ∈K ∪ wy − y +w: w ∈W, y ∈W −K. That is,⟨w: w ∈W |R⟩

(2)

is an abelian group presentation ofI ⊗W L. Now, fix some elementz ∈W −K. For anyk ∈ K, k = e, we have a relatorzk − z + k in the presentation (2). Thus, by applyingseries of Tietze transformations to (2), we see that⟨

k: k ∈K∪ z|k1k2 − k1 − k2: k1, k2 ∈K

∪ y2: y ∈W −K

⟩is also an abelian group presentation ofI ⊗W L. Now, under this identification, thmultiplication mapI ⊗W L→ L sendsk → 0 andz → 2. It is clear now that the kernel othis map can be described as claimed.


We remark that is not hard to prove Lemma 6.3 by a direct argument at the singular

pletede Prof.nt. The

ents.

oof ofrhood

chain level.

Acknowledgements

The work presented here forms part of a Ph.D. thesis [2] which the author comunder the supervision of Prof. George Peschke. The author would like to acknowledgPeschke’s contribution to this work and to thank him for his ideas and encouragemeauthor would also like to thank the referee for several helpful suggestions and comm

Appendix A. Proof of Lemma 6.1

The purpose of this appendix is to provide the details that were omitted from the prLemma 6.1 that was given in the main text. That is, we must show that the neighbouU has properties (1) and (2) as in the statement of the lemma.

Let n := (x0, . . . , xn) ∈ Rn+1: xi 0,

∑xi = 1 be the standardn-simplex. For

x = (x0, . . . , xn) ∈ n andm n, define|x|m := x0 + · · · + xm. Now for all x ∈ n suchthat|x|m = 0 define(m)x ∈ n by

(m)x := 1

|x|m (x0, . . . , xm,0, . . . ,0).

Similarly, for all x ∈ n such that|x|m = 1 definex(m) ∈n by

x(m) := 1

1− |x|m (0, . . . ,0, xm+1, . . . , xn).

We observe that if 0< |x|m < 1, then the points(m)x, x and x(m) are collinear, sincex = (|x|m)(m)x + (1− |x|m)x(m). We can identifym with the spacex ∈ n: |x|m = 1.Let Om := x ∈n: |x|m > 1/2. Thus, there is an inclusion of pairs

im : (n,m) → (n,Om

)whereOm is the closure ofOm in n.

Lemma A.1. Letgm : (n,Om)→ (n,m) be given by

gm(x) :=

x if |x|m = 0,

(2|x|m)(m)x + (1− 2|x|m)x(m) if 0< |x|m < 12,

(m)x if |x|m 12.

Thengm andim are mutually inverse homotopy equivalences.

Proof. The required homotopyH : [0,1] × n →n is given by

Hm(t,x) :=

x if |x|m = 0,( 22−t

|x|m)(m)

x + (1− 2

2−t|x|m

)x(m) if 0 < |x|m < 1− t

2,

(m)x if |x|m 1− t2.

Observe thatHm(t,x)= x for all x ∈m.


Now, using the same notation as in the proof of Lemma 6.1, we have a homeomorphism

ulation

oin

of

t

me.

n

[3],

Berlin,

hesis,

erlin,

graphs,

(1985)

l. 91,ezaa,. Waner.

ft :n → t that mapsm to s. Definegs,t : (t,Os,t) → (t, s) to beft gm f−1t and

Hs,t : [0,1] × t → t to beft Hm (Id[0,1] × f−1t ).

Next, we observe that, as a consequence of the definition of an equivariant triang(Definition 2.5 of [5]), if t is anyn-simplex ofL, thent ∩K consists of asinglem-simplexfor somem n. Thus we can attempt to piece together all the mapsgs,t : (t,Os,t)→ (t, s)

wheres = t ∩K to obtain a single mapg : (L,O)→ (L,K). If s = t ∩K is empty, then welet gs,t : t → t be the identity map. We need only check that ift1 andt2 are simplices ofLwith si := ti ∩K, then the mapsgs1,t1 andgs2,t2 agree ont1∩ t2. This is left to the reader tverify. Similarly, the homotopiesHs,t : t ⊂ L, s = t ∩K can be pieced together to obtaa homotopyH : [0,1] × L → L which shows thatg is in fact a homotopy equivalencepairs.

Of course, to prove thatU has property (1) from Lemma 6.1 we must show thag,respectivelyH , can be lifted to equivariant mapsg : (SW ,U) → (SW ,A), respectivelyH : [0,1]× (SW,U)→ (SW ,A). Again, we can construct these lifts one simplex at a tiLet n(G;K0, . . . ,Kn) denote the standard equivariantn-simplex of type(K0, . . . ,Kn)

(see [5] for the appropriate definition). LetB be the subspace ofn(G;K0, . . . ,Kn)

consisting of the nonfree orbits. Ifπ :n(G;K0, . . . ,Kn)→n is the quotient map, theπ(B)=m for somem n. It is now easy to see that the mapsgm andHm can be lifted toequivariant mapsgm : (n(G;K0, . . . ,Kn),π

−1(Om)) → (n(G;K0, . . . ,Kn),B) andHm : [0,1] × n(G;K0, . . . ,Kn)→n(G;K0, . . . ,Kn) as required.

To see thatU has property (2) from Lemma 6.1 we observe that, by Chapter I ofSW −A is an open dense subset ofSW . ThusSW −A is aG-invariant submanifold ofSW .Let M := SW − U and suppose thatm ∈ M ∩ U . Clearlym /∈ A, som has aG-invariantopen neighbourhoodN together with an equivariant diffeomorphismψ :N → G × R

k .Moreover, it is clear from the construction ofU that we can chooseψ so thatN ∩ M =ψ−1(G× (Rk−1 ×R

0)) as required.

References

[1] M. Crabb, I. James, Fibrewise Homotopy Theory, in: Monographs in Mathematics, Springer-Verlag,1998.

[2] J. Cruickshank, Twisted cobordism and its relationship to equivariant homotopy theory, Ph.D. TDepartment of Mathematical Sciences, University of Alberta, 1999.

[3] T. tom Dieck, Transformation Groups, in: Stud. in Math., Vol. 8, de Gruyter, Berlin, 1987.[4] D. Husemoller, Fibre Bundles, 3rd Edition, in: Graduate Texts in Math., Vol. 20, Springer-Verlag, B

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applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000) 129–183.[6] S. Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, in: Fields Instit. Mono

Vol. 7, American Mathematical Society, Providence, RI, 1996.[7] L.G. Lewis Jr, Open maps, colimits, and a convenient category of fibre spaces, Topology Appl. 19 (1)

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