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Annals of Mathematics

On Manifolds Homeomorphic to the 7-SphereAuthor(s): John MilnorSource: The Annals of Mathematics, Second Series, Vol. 64, No. 2 (Sep., 1956), pp. 399-405Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969983 .

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ANNALS OF MATHEMATICS

Vol. 64,No. 2,September, 956Printed n U.S.A.

ON MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE

BY JOHN MILNORI

(Received June 14, 1956)

The object of this note will be to show that the 7-spherepossesses severaldistinct ifferentiabletructures.

In ?1 an invariantX s constructedororiented, ifferentiable-manifoldsM7satisfyinghehypothesis *) H3(M7) = H4(M7) = 0. (Integercoefficientsreto be understood.)n ?2 a general riterions givenforproving hatan n-mani-fold is homeomorphic o the sphere S'. Some examples of 7-manifolds restudied n ?3 (namely 3-spherebundles over the 4-sphere).The resultsof the

preceding wo sections are used to show that certain of these manifolds retopological -spheres, ut not differentiable-spheres. everal related problemsare studied n ?4.

All manifolds onsidered,with or withoutboundary, re to be differentiable,orientable and compact. The word differentiableill mean differentiablefclass Ca. A closedmanifoldM' is orientedf one generator u Hn(M') is dis-tinguished.

?1. The invariantX(M7)

For every closed,oriented -manifoldatisfying *) we will define residueclass X(M7) modulo 7. According o Thom [5] everyclosed 7-manifoldM7 isthe boundaryof an 8-manifold 8. The invariantX(M7) will be defined s afunction f the indexr and thePontrjagin lasspi ofB8.

An orientation e H8(B8,M7) is determined y the relation v = j.. Defineaquadratic form verthegroupH4(B8,M7)/(torsion)bytheformula -* (v,a 2).

Let r(B8) be the index of this form the number of positivetermsminus thenumber fnegative erms,whentheform s diagonalized ver therealnumbers).

Let pi EH4(B8) be the firstPontrjagin class of the tangent bundle of B8.

(For the definition fPontrjaginclasses see [2] or [6].) The hypothesis *) im-plies thattheinclusionhomomorphism

i:H4(B') M7) -> H4(B8)

is an isomorphism.hereforewe can define "Pontrjaginnumber"

q(B8) - (v, (F-1pi)2).

THEOREM 1. The residue lass of 2q(B8) - r(B8) modulo does not dependonthe hoice fthemanifold 8.

DefineX(M7)as this residueclass.2As an immediate onsequencewe have:COROLLARY. If X(M7) F 0 thenM7 is nottheboundary f any 8-manifold

having ourth ettinumber ero.

The author holds an Alfred P. Sloan fellowship.

399

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400 JOHN MILNOR

Let B', B" be twomanifoldswith boundaryM. (We may assume they aredisjoint.) Then C8 =_B8u B8 is a closed 8-manifoldwhich possessesa differ-entiable structure ompatiblewith that ofBi and B2 . Choose that orientationv

for C8 which s consistentwith the orientationPof

B8(and therefore on-sistentwith v2). Let q(C8) denote the Pontrjaginnumber v, p2(C8)).

According o Thorn 5] or Hirzebruch 2] we have

r(C8) = ( (7p2(C8) -2 )

and therefore

45r(C8) + q(C8) = 7(v, p2(C8)) 0 (mod 7).

This implies

(1) 2q(C8) - --(C8) 0 (mod 7).LEMMA 1. Under he bove onditions e have

(2) (C0) = (B)- ((B8)

and

(3) q(C8) = q(B8) -q(B ).

Formulas 1, 2, 3 clearly mply hat

2q(B 8)- r(B ) 2q(B8) - r(B ) (m d 7);which s just theassertion f Theorem1.

PROOF OF LEMMA 1. Considerthe diagram

Hn(B1, M) 03 HN(B2,M) h- Hn(C, M)

lil G i2

Hn(B1) 0 Hn(B2) <k Hn(C)

Note that for n = 4, these homomorphisms re all isomorphisms. fa = jh-1(a, 0 a2) e H14(C), then

2) j- 2 ($ 2)) = (V1 E3 (V) Y2 d3s2) =(, t) v2, )4) (v, a2 =P (vjh~(a 0 2~X =PP 2), aG a~ (Pi, a') - 2, .

Thus the quadratic form fC8 s the "direct sum" ofthe quadratic form fB7and thenegativeof the quadraticform fB . This clearly mpliesformula2).

Define , = i-T1p1(B1)nd a2 = ijlp(B2). Then the relation

k(p1(C)) = pi(Bi) 03 p1(B2)

implies hat

2 Similarly forn = 4k - 1 a residue class X(Mn) modulo SkM(Lk) could be defined. (See

121 age 14.) For k = 1, 2, 3, 4 we have SkY(Lk) = 1, 7, 62, 381 respectively.

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MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE 401

jh-1 a, 0) a2) = PI(C).

The computation 4) nowshows that

(V,P1(C)) = (VI,Ia')-

(V2 ),which s just formula 3). This completes heproofof Theorem 1.

The following roperty f theinvariantX s clear.LEMMA 2. If theorientationfM7 is reversedhenX(M7) is multiplied y -1.As a consequencewe haveCOROLLARY. If X(M7) 5 0 thenM17 ossessesno orientationeversingiffeo-

morphism3nto tself.

?2. A partial haracterization fthe n-sphere

Consider hefollowing ypothesis oncerning closedmanifoldMn whereRdenotes herealnumbers).(H) There existsa differentiableunction :Mn -* R havingonlytwocritical

points o x1 Furthermorehese ritical oints renon-degenerate.(That is if u1, , un are local coordinates in a neighborhood of xo (or x1)

thenthematrix a2f/aupauj)s non-singulart xo (orxi).)THEOREM 2. If M' satisfieshehypothesisH) then hese xists homeomorphism

ofM' onto ' whichs a diffeomorphismxcept ossiblyt a singlepoint.Added n proof. his result s essentially ue to Reeb [7].

The proofwill be based on the orthogonal rajectories f themanifolds =constant.Normalizethe function so thatf(xo) = 0, f(xi) = 1. According o Morse

([3] Lemma 4) there exist local coordinates vl, n, in a neighborhood V ofxoso thatf(x) = 2l+ + vnfor x e V. (Morse assumesthatf s of class C3,and constructs oordinatesof class C1; but the same proofworks in the C'case.) The expression s2= dvl+ . + dv2defines Riemannian metric ntheneighborhood . Choose a differentiableiemannianmetricforMnwhichcoincideswiththis one in someneighborhood4' ofxo Now thegradien off

can be considered s a contravariant ectorfield.FollowingMorsewe consider he differentialquation

dx = grad fIl 1rad f 112.dt

In theneighborhood ' thisequationhas solutions

(vi t), v, (t) ) = (a, (t) ,***,an (t))

for0 ? t < c, where a = (al, ***, an) is any n-tuple with a = 1. Thesecan be extendeduniquely o solutions <(t)for0 < t ? 1. Note thatthesesolu-tionssatisfy he dentity

3Adiffeomorphismfs a homeomorphismnto, uchthatbothf ndf'I aredifferentiable.4This is possible by [4] 6.7 and 12.2.

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402 JOHN MILNOR

f(xa(t)) = t.

Map the nterior f theunitsphereofR' intoM' by themap

(al(t)',

* X. antWIt is easilyverified hat thisdefines diffeomorphismf the open n-cellonto

-n - (xi). The assertion fTheorem2 nowfollows.Givenanydiffeomorphism: S8'--> AS", an n-manifoldan be obtainedas

follows.CONSTRUCTION (C). Let M (g) be themanifold btainedrom wo opiesofRn

bymatchinghe ubsets ' - (0) under hediffeomorphism

1 _u_

U * V= 9

(Such a manifolds clearlyhomeomorphico Sn. If g is the identitymap thenMn(g) is diffeomorphico S'.)

COROLLARY 3. A manifoldMB can be obtained y theconstructionC) if and

only f itsatisfieshehypothesisH).PROOF. fMln(g) is obtainedbytheconstructionC) thenthefunction

f(x) u 127(1 + 11 112) 1/(1 + 11112)

will satisfy he hypothesis H). The conversecan be establishedby a slight

modificationf the proof fTheorem2.?3. Examples of 7-manifolds

Consider3-spherebundlesover the 4-spherewiththe rotationgroupS0(4)

as structuralgroup. The equivalence classes of such bundles are in one-one

correspondence5 ith elementsof the group 7r3(SO(4)) Z + Z. A specificisomorphismetween hesegroups s obtained s follows. or each (h, ) e Z + Z

letfhj:S3 * SO(4) be defined y fhj(u)-v = UhvUi,forv e R4.Quaternionmultipli-

cation s understood n theright.

Let l be the standard generator orH4(S4). Let thj denotethe spherebundlecorrespondingo (fhj) e 2-3(SO(4)).LEMMA 3. ThePontrjagin lassPm(thj) equals ? 2(h - j)(The proofwill be given later. One can show that the characteristic lass

C(hj) (see [4]) is equal to (h + j) .)For each odd integerk let M7 be the total space of the bundle hjwhere h

and j are determined y the equationsh + j = 1, h - j = k. This manifld

Mk has a natural differentiabletructure nd orientation,whichwill be de-

scribed ater.

LEMMA 4. The nvariant (M7) is the esidue lass modulo ofk21.

LEMMA 5. Themanifold 7 satisfieshehypothesisH).Combining hesewehave:

5See [4]?18.

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MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE 403

THEOREM 3. For k2 A 1 mod 7 themanifold k is homeomorphicoS7 butnotdiffeomorphico 7.

(For k = ? 1 the manifoldM7 is diffeomorphico S7; but it is not known

whether his s truefor ny otherk.)Clearly ny differentiabletructure n S7 can be extended hroughR8 - (0).However:

COROLLARY 4. There xists differentiabletructuren S7 which annotbe ex-tended hroughout8.

This followsmmediately rom he preceding ssertions, ogetherwithCorol-lary1.

PROOF OF LEMMA 3. It is clear that the Pontrjaginclass Pl(thj) is a linearfunction fh and . Furthermoretisknown hat t is independent fthe orienta-

tionof the fibre. utif the orientation f S3 is reversed, hen hj is replacedby

t-j-h. Thisshows hat l(thj) is given yan expressionfthe form(h - j)t.Herec is a constantwhichwillbe evaluated later.

PROOF OF LEMMA 4. Associatedwith ach 3-sphere undleMk *S4 there s a4-cell bundlePk -_ S4. The total space B8 ofthis bundleis a differentiablemanifoldwithboundary 7 . The cohomology roupH4(Bk) is generated y theelement = pk t). Choose orientations ,vforMk and B8 so that

(V, (i10)2) = +1.

Thenthe ndexr(Bk) will be + 1.The tangentbundle ofBk is the "Whitney um" of (1) thebundleof vectors

tangent o the fibre,nd (2) the bundle ofvectorsnormal o the fibre. he firstbundle 1) is induced under k) fromhebundle hj, and thereforeasPontrjaginclassPi = pk c(h - j)) = ckla.The second s inducedfrom hetangentbundleof 4, and thereforeas first ontrjagin lass zero.Now bytheWhitney roducttheorem [2] or [6])

p1(B = cka + 0.

For thespecial case k = 1 it is easilyverified hatB8 is the quaternionpro-jective planeP2(K) withan 8-cell removed.But thePontrjagin lass pl(P2(K))is known o be twice a generator fH4(P2(K)). (See Hirzebruch 1].) Thereforetheconstant mustbe 4?2,whichcompletes heproof f Lemma 3.

Nowq(B8) = (v, (A1(?-2kl))a) = 4k2; nd 2q - T = 8k2 1 2 - 1(mod7). This completes heproof f Lemma 4.

PROOF OF LEMMA 5. As coordinateneighborhoodsn the base space S4 takethecomplementfthenorthpole,and thecomplementfthe southpole. Thesecan be identified ith euclideanspace R4understereographicrojection.Thena pointwhichcorresponds o u ER4 under one projectionwill correspond o

u = u/l u 112under theother.The total space M7can now be obtained as follows.5 ake two copies of

R4(X S3 and identifyhesubsets R4- (0)) X S3 underthediffeomorphism

(u, v) (u', v') = (U/|| u ||, uvu /11 | |l)

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404 JOHN MILNOR

(using quaternion multiplication).This makes the differentiabletructure fMkprecise.

Replace the coordinates u', v') by (u", v') whereu" = u'(v')-'. Considerthefunction:M7 R defined

yf(x) ?(v)/(1 11 lI,)! = 9(u")/(l + 11 1II2);

where9(v) denotes thereal part of the quaternionv. It is easily verified hat fhas only two criticalpoints namely u, v) = (0, ?i1)) and that these are non-degenerate. his completes he proof.

?4. Miscellaneous results

THEOREM 4. Either a) there xists closed opological -manifold hich oesnotpossess ny differentiabletructure;r b) the ontrjagin lasspi of n open8-mani-fold s not topologicalnvariant.

(The authorhas no dea which lternative olds.)PROOF. Let X8 be the topological8-manifold btainedfromBk by collapsing

its boundary a topological7-sphere)to a pointxo. Let a eH4(X ) correspondto thegenerator EH4(B ). SupposethatX8, possesses differentiabletructure,and that p1(X8 - (xo)) is a topological nvariant.Then pl(X8) must equal+2ka, hence

2q X8) -r(X8) = 8k2 _ 1- k2 - 1 (mod 7).

But fork2 1 (mod7) this s impossible.Two diffeomorphisms, g:M' -* M' will be called differentiablysotopic f

there xists diffeomorphism' X R-> M' X R oftheformx,t) * (h(x, t),t)suchthat

h _xff x (t ?< 0)h(x t) - fg(x) (t > 1).

LEMMA 6. If thediffeomorphisms, g:Sn- _> S-1 are differentiablysotopic,then hemanifoldsM'(f), M'(g) obtained ythe onstructionC) arediffeomorphic.

The proof s straightforward.THEOREM 5. There exists diffeomorphism:S'6 > S6 of degree 1 which s

notdifferentiablysotopic othe dentity.Proof. By Lemma 5 and Corollary3 the manifoldM' is diffeomorphico

17(f) for omef. Iff weredifferentiablysotopicto theidentity henLemma 6would mplythatM' was diffeomorphico S7. But this s falseby Lemma 4.

PRINCETON UNIVERSITY

REFERENCES

1. F. HIRZEBRUCH, Ueber ie quaternionalenrojektivendume, .-Ber. math.-naturw.KI.Bayer.Akad. Wiss. Mifnchen1953),pp. 301-312.

2. , Neue topologischeMethoden n deralgebraischenGeometrie, erlin,1956.

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MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE 405

3. M. MORSE,Relationsbetween henumbers fcriticalpointsofa real function f n inde-pendent ariables, rans. Amer.Math. Soc., 27 (1925),pp. 345-396.

4. N. STEENROD, The topology f fibre undles,Princeton, 951.5. R. THOM,Quelquesproprietes lobale esvarietes ifferentiables,omment.Math.Helv.,

28 (1954), pp. 17-86.6. Wu WEN-TSUN,Sur les classescaracteristiqueses structuresibrees pheriques, ctual.sci. industr. 183,Paris, 1952, p. 5-89.

7. G. REEB, Sur certain ropri6tesopologiquesesvarieteseuilletees, ctual. sci. industr.1183, aris, 1952, p. 91-154.