A Manifold which does not admit any Differentiable Structure by MICHEL A. K~RVXm~, New York (USA) An example of a triangulable closed manifold M o of dimension 10 will be constructed. It will be shown t h a t M o does not admit any differentiable structure. Actually, M o does not have the homotopy type of any differentiable manifold. Also, a 9-dimensional differentiable manifold 2:9 is obtained. 2:9 is homeomorphic but not diffeomorphic to the standard 9-sphere S 9. Use is made of a procedure for killing the homotopy groups of differentiable manifolds studied by J. MIL~OR in [6]. I am indebted to J. MIL~OR for sending me a copy of the manuscript of his paper. Although much of the constructions (in particular the construction of M0) generalizes to higher dimensions, I did not succeed disproving the existence of a differentiable structure on the higher dimensional analogues of M 0. A more general case of some of the constructions below will be published in a subsequent paper, with other applications. 1) w 1. C o n s t r u c t i o n o f a n i n v a r i a n t

Let M 1~be a closed triangulable manifold. Assume t h a t M 1~is 4-connected. (M 1~is connected, and ~i(M) = 0 for 1 ~ i --< 4.) I t follows from Por~cxR~ duality and the universal coefficient theorem t h a t H a ( M ; G ) ~ 0 for 5 < q < 10, and H S ( M ) is free abelian of even rank 2s, say. (If no coefficients are mentionned, integer coefficients are understood.) Let Q = ~ S 6 be the loop-space on the 6-sphere. I t is well known t h a t H 5 ( D ) = Z, H l ~ and if ~ : ~ x D - ~ is the map given by the product of loops, then n*(el) = e 1|

1 ~- l |

y~* (e2) ~--- e 2 @ 1 +

1 |

1,

and

e 2 ~- e 1 |

el,

where el, e2 are the generators of Hs(Q) and Hl~ respectively, and H* (D • Q) is identified with H * ( ~ ) | H* (D) by the K ~ E T H formula. (Compare R. BOTT and H. S.~M~LSO~ [1], Theorem 3.1 .B.) L e m m a 1.1. Let X c H s(M)

be given. There exists a map [ : M ~ ~2 such

that /*(e~) = X . 1) This paper was presented at the International Colloquium on Differential Geometry and Topology, Zurich, June 1960.

258

Mmm~. A. K ~ v ~

Proo/. L e t K be a triangulation of M . Define / by stepwise extension on the skeletons K (q) using obstruction theory. /I K(4) is taken to be the constant map into a base point o n / ~ . Let Xo be a representative cocycle of X . For every 5-dimensional simplex s 5 of K , define / I s5 to be a representative of X0[ss]-times the generator of gs(~) _--__:~e(S 6) c~ Z. The obstruction cocycle to extend / [ K (s) in dimension 6 is zero. The next obstruction is in dimension 10 with values in g9(/2) ~ ~10(S e) -----0. (See [9], w41.) Thus the lemma is proven. Define a function ~o: H s ( M ) --> Z2 b y the following device. For every X ~HS(M), take a map f: M - > Q such that f * ( e i ) ~--- X . Then, ~0(X) = = / * (u2) [M], where u 2 ~ H Ie (~-2 ; Z~) is the reduction modulo 2 of e~ c H l~ (Q), and /* (u2) [M] is the value of the cohomology class /* (u2) on the generator of H10(Ml~ Z2). Lemma 1.2. The function ~o: H s ( M ) --> Z2 is well defined, i.e., qo(X) does not depend on the choice o/ the map / : M --+ s such that /* (ex) = X . Proof. Let f , g : M--->12 be two maps such that /*(el)-----g*(el). We have to show that f* (u~) = g* (u2). Let K again be a triangulation of M . Since / * ( e l ) - g * ( e l ) = 0, it follows that / and g are 5-homotopic. (See S. T. H v [2], Chap. VI.) Since H a ( M ; gq(%))) ---- 0 for 5 < q < 10, it follows that f and g are 9-homotopic. Hence, we may assume that [I K(9) : g I Keg). Let 0)10(/, g) E C 10( K ; ~10(/J)) be the difference cochain. Then, (f* (U2)

-

-

g~< (Us)) [810 ] = U 2 [h (Dl~

g) [Sio]] ,

for every 10-simplex Sl0 , where h: ~Zl0(~) --> H10(~ ) is the I-Iu~,wIcz homomorphism. According to J. P. Sv.RRE, u 2 [h~] is the rood. 2 H o P r invariant of the element in g11(S 6) represented b y ~ e g~0(%)S~). (Compare [8], Lemme 2.) Since no element of odd HoP~ invariant occurs in ~11(S~), it follows that /* (u~) ~- g* (u~), and the proof is complete. Lemma 1.3. Let X , Y ~ H~(M) be two integer cohomology classes of M . Then, ~o(X + Y) = ~o(X) + ~0(Y) + x . y , where x . y is the value on the generator of H10(Ml~ Z2) of the cup-product x ~ y. (x, y are the rood. 2 reductions of X and Y respectively.) Proo]. Let f, g: M - - , 1) be maps such that / * ( e l ) = X and g * ( e l ) = Y . B y definition, q0 (X) = / * (u~) [M], and ~0 (Y) =- g* (u2) [M]. Let / x g : M • ~1)• be the product o f / and g. (I.e., / x g ( u , v) = = (f(u), g(v)).) Let D : M ~ M • M be the diagonal map. Define 1~ : M - * ~

A Manifold which does not admit any Differentiable Structure

259

by F ~ o ( [ • g) o D , where z~: ~ • ~ - ~ is given by the multiplication of loops. Since D* maps the tensor product of cohomology classes into their cup-product, we have F* (el) --~ D* (X | 1 ~- 1 | Y) ~ X ~- Y. Therefore, ~0(x + Y) = ~* (u~) [ M ] . On the other hand,

F* (u2) = D* if* (u2) | 1 + 1 | g* (u~) + / * (ul) | g* (ul)) : /* (U2) -~ ~$ (U2) -~- /$ (Ul) ~.1g$ (Ul) = / * ( u 2 ) + g*(u2) + x ~ y . (u 1 is the reduction modulo 2 of eI .) This proves Lemma 1.3. The function ~0 : H6 (M) --> Z 2 induces a function ~ : H 5(M ; Z2) -~ Z~. satisfying ~(x ~- y) = ~(x) + ~(y) ~- x . y . Indeed, if X is an integer class whose reduction modulo 2 yields x E H ~ ( M ; Z~), we define ~o(x) -~ Co(X). It follows from ~0(2 Y) ---- ~0(Y) ~- ~o(Y) + y ' y = y ' y -~ 0 , t h a t ~(x) ~ Z2 depends only on x e l l S ( M ; Z~). The function ~ : H a ( M ; Z2) --> Z~ is then used to construct the number (M) as follows. A basis x l , . . . , x a, y l , . . . , y, of H s(M; Z2) as a vector space over Z2 will be called 8ymplectic if x~. x~ --~ Yi'Y~ ~ O, and x i . y j ~ Oin for all i, ?"-----1 , . . . , s. Clearly, symplectic bases always exist. Moreover, it is well known t h a t since the function ~ : H 5(M; Z2) -~ Z2 satisfies the equation ~ ( x + y) : q~(x) -~ ~(y) ~- x . y , the remainder modulo 2 r

= ,Y,~qJ(x,).q~(y,)

is independent of the symplectic basis x ~ , . . . , x~, y~ . . . . , y,. The rest of the paper is devoted to investigating the properties of the invariant ~ . Clearly, r is an invariant of the homotopy type of 4-connected closed manifolds of dimension 10. Our objective is the proof of the following theorems. T h e o r e m 1. I / M ~~ has the homotopy type o / a C~-diHerentiable 4-connected closed mani/old, then r (M) -----O. ( i t can be shown t h a t the converse of this theorem would follow from the conjecture t h a t the cohomology ring H* (M) and ~b(M) are a complete set of invariants of the homotopy type of the triangulable 4-connected closed manifold M of dimension 10.)

260

Mm~L

A.

K~vJm~E

Theorem 2. There exists a closed 4-connected combinatorial manifold M o of dimension 10 for which r : 1. (In fact a specific example will be constructed.) In w2, the proof of Theorem 1 will be carried out taking Lemmas 4.2 and 5.1 for granted. (Lemma 4.2 is used in the proof of Lemma 2.2, and Lemma 5.1 is used to deduce Theorem 1 from Lemma 2.4.) The Lemmas 4.2 and 5.1 are proved at the end of the paper, in w4 and w 5. Theorem 2 will be proved in w 3. w 9. Proof of Theorem 1 Let M TMbe a closed Cl-differentiable manifold which is 4-connected. Lemma 9.1. M TM is a re-manifold. Proof. Let M l ~ R ~+1~ be an imbedding with n large. We have to show that the normal bundle v is trivial. This is done b y constructing a field of normal n-frames f,. Let K be a triangulation of M ~~ Since n, ($0,) ~ 0, and M 1~ is 4-connected, it follows that Hq+l ( M ; ~q (S On)) = 0 for 0 ~ q < 9. Thus, there is only one possibly non-vanishing obstruction o(~, f,)~ H I ~ ~ 9 ( 3 0 , ) ) ~ ~9(30~) to the construction of the field f, of normal n-frames. B y Lemma 1 of [7], o(v, f~) is in the kernel of the HoPFWHrrEHEXD homomorphism J0 : ~9($0,) -~ ~n+9(S~). But J9 is a monomorphism. (Compare proof of Lemma 1.2 of [4].) Hence, o(v, f~) : 0, and the lemma is proved. (Recall that the proof of the assertion: J9 is a monomorphism, was based on Corollary 2.6 of J. F. ADx~s paper On the structure and ap?lications of the STEE~OD algebra, Comm. Math. Helv. 32 (1958), 180-214. This statement also follows from the portion of the POST~IKOV decomposition mod. 2 of S n given below in w 5.) The THOM construction associates with every framed manifold (M; f~), where M C R "§ an element ~ ( M ; f~) r We say that (Ml~ f,) is homotopic to zero if the corresponding element ~ ( M ; f~) is the neutral element of ~,+10(Sn) 9 Lemma 2.2. If then r -~ O.

(M~O; f~) is homotopic to zero, where M ~~ is 4-connected,

Proof. The assumption that (M; [~) is homotopie to zero implies the existence of a framed manifold (Vll;F~) with boundary M 1~ (Compare R. THoM [10].) We may assume that V is connected, and hence has a trivial tangent bundle. We can therefore apply to V -- M the procedure for killing the homotopy groups of a differentiable manifold studied b y J. MrrxoR. Specifically, using Theorem 3 of [6], we obtain a new ll-dimensional differen-

A Manifold w h i c h does n o t a d m i t a n y Differentiable S t r u c t u r e

261

tiable manifold with b o u n d a r y M t0 which is also 4-connected. This new 4connected manifold will again be d e n o t e d b y V 1~. W e can now forget a b o u t the fields of normal frames. W e proceed to compute ~5(M). Consider t h e cohomology e x a c t sequence of the pair (V, M) with coefficients in Zs,

9 .. ---~H6(V)--).HS(M)-~H6(V, M) --~ . . . . Using relative POII~CAR~-LEFSCHETZ d u a l i t y (over Zs), and the formula

u ~ ~x[V, M] = i*(u) ~ x[M] , where u e H s(V), x e H 5(M) and IV, M ] , [M] are the generators of Hlt(V,M;Z2) and H l o ( M ; Z s ) respectively, it follows t h a t H S ( M ; Z s ) has a symplectic basis x~ . . . . . xs, Yt . . . . , y8 say, such t h a t x l , . . . , x 8 is a v e c t o r basis of K e r ~. Consequently, in order to p r o v e r ~ 0, it is sufficient to show t h a t ~ (x) : 0 for e v e r y x c K e r 8. L e t X ~ H 5 (M) be an integer class whose reduction modulo 2 is x, a n d let [: M 1~ - ~ ~ f2S 6 be a m a p such t h a t /*(el) : X . We have to show t h a t [*(u2) : 0, where u~ generates H~~ Zs). L e t ~ * be t h e space o b t a i n e d from D b y attaching a cell of dimension 6 b y a m a p S 5 -~ ~ of degree 2. B y L e m m a 4.2 in w 4, below, for e v e r y m a p g : S 1~ -~ Q*, one has g* (u2) ---- 0, where we denote b y u 2 E H t~ ; Zs) again the class corresponding to the old Us cH~~ Z2) u n d e r the canonical isomorphism H1o(~2 ; Z2) ~ H I ~ Z~). We a t t e m p t to e x t e n d [ : M - ~ 1 2 * to a m a p of V i n t o ~ * . L e t ( K , L ) be a triangulation of (V, M ) . The stepwise extension of [ on t h e skeletons K(q) ~ L leads to obstructions in the groups H q + ~ ( K , L ; r c q ( ~ * ) ) . F o r q < 5, gq(~2*) ---- 0. We meet a first obstruction for q = 5 in H 6 ( K , L ; Z2). B y the HoPF theorem, this obstruction is ~x. (See S. T. H v [2].) Since Ox --~ 0, it is possible to e x t e n d / on K ( 8 ) ~ L . Using H q + ~ ( K , L ; G ) - ~ O for 5 < q < 10 (since V is 4-connected), it follows t h a t there exists a m a p F : K -- ~ -+ Q*, where ~ is some ll-dimensional simplex in K - - L , such that F I L ~/. L e t S 1~ denote the b o u n d a r y of ~, a n d let g : S 1~ --~ D* be the restriction of F on S 1~ Since O(K -- v) -----L -- S 1~ and g*(us)-~O, it follows t h a t [* (us) ~ 0. The proof of L e m m a 2.2 is complete. Corollary 2.3. I f two 4-connected /tamed mani/olds ( M ; f~) and ( M ' ; f~) o~ dimension 10 de/ine the same element ~ - - - - ~ ( M ; f n ) - ~ ( M ' ; f~) by the THOM ConStruction, then r = ~)(M'). This is o b t a i n e d b y observing t h a t r is additive with respect t o the c o n n e c t e d sum of manifolds. I t follows t h a t r provides a h o m o m o r p h i s m from a subgroup of ~.+t0(S ~)

262

~ h c ~ L A. K E ~ v ~

into Z2. We denote this homomorphism b y ~ again. Actually, ~ is defined on every element of ~n+lo(S~). Indeed, using spherical modifications [6], it is easy to see that every element ~ E ~ . + 1 o ( S " ) is obtainable from a 4-connected framed manifold b y the THOM construction. This remark will not be used in the present paper. I t follows from Corollary 2.3 that Theorem 1 is equivalent to the statement that # ( ~ ) = 0 for every ~ c~n+lo(S=), provided q}(~) is defined. Since q~(~) is obviously zero for every element c, of odd order, and b y J. P. SERRE'S results ~t,+10(S=) contains no element of infinite order, it is sufficient to show that r annihilates the 2-component of the group ~n+10(S=) 9 B y Lemma 5.1 in w 5 below, every element ~ in the 2-component of ~n+lo(S ~) is representable in the form ~=/~o~, where ~ ~ ~,,+1o(Sn+9) is the generator of the stable 1-stem, and t5 ~ ~+9(S~). ~ence, Theorem 1 will follow from the Lemma 2.4. Every element ~r c~+~o(S ~) o/ the [orm o~ = fl o T, with E gn+lO(S~+9), and fl E g~+,(S '~) is obtainable by the T~OM construction /rom a framed mani/old (I1~ f~), where 11~ has the homotopy type o/the lO-sphere ~10. Proo/. W e first show that fle ~ + 9 (S~) is obtainable by the T~oM construction from a framed manifold (27~; f~), where 279 has the homotopy type of the 9-sphere. It is well known that fl is obtainable b y the THOM construction from some framed manifold (Mg; f~). We have to show that (Mg; f~) is homotopie to a framed manifold (279; fn), where 279 is a homotopy sphere. This is done b y simplifying M 9 by a series of spherical modifications. (See J. MIL~OI~ [6].) Assuming b y induction that M 9 is (p -- 1)-connected (0 < p ~ 4), we have to prove that (M; f~) is homotopic to a p-connected framed manifold (M'; f~). Recall that a spherical modification of type (p + 1, q + 1) applied to a class 2 E ~ ( M 9) consists of the following construction. Represent 2 b y an imbedding / : S" • Dq+~ --->M ~ ,

with p + q + 1 = 9. (This is possible for p g 4 since M 9 is a ~-manifold and the normal bundle of any imbedding S ~ -+ M ~ is stable in this range of dimensions.) The manifold M is then replaced b y M'=

( M - - I ( S ~ x Dq+,)) ,~ (D,+, x Sq) ,

under identification of /(S = x S q) regarded as the boundary of / ( S ~ x D q+~) with S ~ x Sq regarded as the boundary of D ~+t x Sq. B y Theorem 2 of

A Manifold which does not admit any Differentiable Structure

263

[6], the manifolds M and M' bound a 10-dimensional differentiable manifold ~o ---- ~o(M, f), and / : S ~ • n q+l --> M 9 can be chosen such t h a t the field fn (over M) is extendable over co as a field of normal n-frames. (We can think of co as imbedded in R n+l~ with M c R ~+9• (0) and M' c R ~+9• (1) since n can be taken as large as we please.) Hence spherical modifications of type ( p ~ 1 , q ~ 1) with 0 ~ p _ ~ 4 can be performed so as to carry (M; f,) into a homotopic framed manifold. It is known (Theorem 3 of [6]) that for p < 4, spherical modifications simplify the manifold. More precisely : ~ ( M ' ) is isomorphic to the quotient of g~(M) by the subgroup generated by ~t, and ni(M) c~ z i ( M ' ) ~ 0 for i < p. Hence, it is easy, using [6], to obtain a 3-connected framed manifold homotopie to (Mg; f~). The case p ~ 4 requires special care. If ~ e na(M 9) is the class we want to kill, there exists an imbedding / : S a • D s - + M 9 such t h a t / I S 4 • (0) represents ~. Let M' ~-- x ( M , / ) be the 9-dimensional manifold obtained from M and / by spherical modification. ( / i s supposed to be chosen so t h a t (M'; f~) with some f~ is homotopic to (M; fn).) In general, however, / I x0 x (bdry D 5) represents a non-zero element of na(M'). Thus, it is not clear a priori t h a t a series of spherical modifications of type (5, 5) will carry M into a 4-connected manifold, and hence a homotopy sphere. If ~ is a generator of the free part of na(M) ~ H a(M), there exists by P o I N c x ~ duality a class # ~ H6(M) whose intersection coefficient with (or h ~t rather, where h is the HUREWICZ homomorphism) is 1. It follows t h a t in this case the cycle given by / I Xo • (bdry D 5) is homologous to zero in M--/(Sa• Ds), and hence in M ' . Thus H a ( M ' ) ~ na(M') has strictly smaller rank than H a(M) ~ na(M), and the torsion subgroup is unchanged. I claim t h a t if 2 Ega (M) is a torsion element, the homology class of the cycle /I Xo • (bdry D 5) is o/ in/inite order /or any ~ representing ~. Hence, one more spherical modification will lead to a manifold with 4-dimensional homology group of not bigger rank than Ha(M) and with a strictly smaller torsion subgroup. (I.e., a series of spherical modifications will lead to a 4connected framed manifold homotopic to (Mg; f~). By PomcxR~ duality, a closed 4-connected manifold of dimension 9 has the homotopy type of $9.) Since the BETTI numbers Pa, P~ of M and M' (in dimension 4) differ at most by 1, and differ indeed by 1 if and only if T (represented by / ] x o • (bdry Ds)) in M' is of infinite order, it is sufficient to show t h a t p~ -{- Pa ----- 1 rood. 2. Since p~ ~-- p~ for 0 <: i ~ 3, this is equivalent to showing t h a t the semicharacteristics E * ( M ) and E * ( M ~) of M and M' (over the rationals, say) satisfy E* (M') -]- E* (M) ---- 1 mod. 2. We use the formula

E * ( M ' ) ~- E * ( M ) -~ E(eo) ~- r

rood. 2,

264

MICHELA. KEItvAn~

where E(eo) is the EIrLER characteristic of t h e manifold oJ with b o u n d a r y r M r - - M , and r is the r a n k of the bilinear form on H 5 (oJ, d) ; Q) defined b y the cup-product. (Compare IV[. A. K~RV~RE [3], w 8, formula (8.9).) I t is easily seen t h a t E(co) -----1, u p t o sign, and since u . u ~ 0 for e v e r y u c H s (oJ, eb; Q), t h e r a n k r must be even: r----0 (mod. 2). I~ence, E*(M ~) ~ E * ( M ) ~ . 1 rood. 2. Summarizing, we have p r o v e d so far t h a t e v e r y fl E gn+9(Sn) is obtainable b y the T H o ~ construction from a f r a m e d manifold (279; f~), where the manifold Z 9 has t h e h o m o t o p y t y p e of S 9. Taking a representative / : S~+1~ S ~+9 of ~ such t h a t /-1(Sn+9 --xg) is diffeomorphic to S 1 • (S ~+9 -- x0) , we obtain t h a t ~ -----fl o ~ is obtainable b y the TRoM construction f r o m (S 1 • X~; f~). I t remains to show t h a t (S 1 • 279; f~) is h o m o t o p i c to a f r a m e d manifold (271~ f~), where 2:1~ is a h o m o t o p y sphere. A p p l y once more the spherical modification t h e o r e m s (Theorems 2 and 3 of [6]), this time t o the class 2 c z l ($1 • 279) represented b y S 1 • (z0). The resulting f r a m e d manifold is h o m o t o p i c to (S 1 • 2:9; f~) and has the homot o p y t y p e of the 10-sphere. This completes t h e proof of L e m m a 2.4. T o complete t h e p r o o f of T h e o r e m 1 it remains to prove t h e L e m m a s 4.2, and 5.1. This is done in w 4 and w 5.

w 3. Construction of Mo

This section relies on J . MILNOR's p a p e r [5]. Let differentiable m a p whose h o m o t o p y class (/o) satisfies i,(/o)

=

/o: S t - * S04 be a

ais ,

a : g6(S 6) -* ~4($05) is t a k e n from t h e h o m o t o p y e x a c t sequence of and i : 304 -* S05 is the usual inclusion. Define /1 ---- [2 - - i o/0. Using ]1,/2: S t - ' 3 0 5 , a diffeomorphism / : S 4 • S t - . S t • S t is given b y / ( x , y) : (x', y'), where y' -~/l(x).y, a n d x -~/2(y').x'. L e t M(/1,/2) be the MILNO~ manifold obtained f r o m the disjoint union of D s • S t and S 4 • D 5 b y identifying each point (x, y) in the b o u n d a r y of D 5 x S t with / ( x , y), considered as a point on t h e b o u n d a r y of S 4 • D 5. B y L e m m a 1 of [5], t o g e t h e r with t h e r e m a r k at the b o t t o m of page 963 in the p r o o f of L e m m a 1 in [5], it follows t h a t the differentiable manifold M (/1, ]2) is h o m e o m o r p h i c to the 9-sphere. I t will follow from T h e o r e m 1 in this paper, t h a t M (/1,/~) is not diffeomorphic to t h e s t a n d a r d S 9. L e t W 1~ be t h e differentiable maniwhere

SOe/S05,

A M a n i f o l d w h i c h does n o t a d m i t a n y D i f f e r e n t i a b l e S t r u c t u r e

265

fold with b o u n d a r y M (fl, f2) o b t a i n e d using the c o n s t r u c t i o n on p a g e 964 of [5]. W can a l t e r n a t e l y be described as follows. L e t U be a t u b u l a r neighborh o o d of t h e diagonal A in S 5 • S 5. I t is well k n o w n t h a t U is the s p a c e of t h e fibre bundle p : U -~ S 5 with fibre D 5 associated w i t h the t a n g e n t b u n d l e of S 5. T h e differentiable m a n i f o l d W is o b t a i n e d b y s t r a i g h t e n i n g t h e angles of t h e quotient space of t h e disjoint union of two copies U ~ a n d U" of U u n d e r a n identification of p ' - l ( V ) with p~-l(V) such t h a t t h e images of A' a n d A" in W h a v e intersection n u m b e r 1. ( V is a n i m b e d d e d 5-disc on S s, and p ' - I ( V ) ~ D 5• D s is identified with p " - I ( V ) ~ D 5• D 5 u n d e r (u, v) ~-~ (v, u), u , v E n 5.) Since W is a 10-dimensional manifold whose b o u n d a r y M ( f l , / 2 ) is homeomorphic t o S 9, t h e union of W with t h e cone o v e r the b o u n d a r y is a 10-dimensional closed m a n i f o l d M o, Since M (fl,/2) is combinatorially e q u i v a l e n t to S g, it follows t h a t M 0 possesses a combinatorial structure. (Compare J . MmNoR, On

the relationship between digerentiable manifolds and combinatorial manifolds, m i m e o g r a p h e d notes 1956, w4.) I t is easily seen t h a t M 0 is 4-connected. W e proceed to c o m p u t e r (M0). L e t x, y c H a (M 0 ; Z~) be t h e c o h o m o l o g y classes d u a l to t h e h o m o l o g y classes of t h e i m b e d d e d spheres ~', ?'" : S 5 -~ M 0 given b y t h e images in W of the diagonals A ~ a n d A" in U r a n d U" respectively. Clearly, x, y is a symplectic basis of H 5 (M 0 ; Z~). (I. e., x- x --~ y- y ~ 0, a n d x . y ~- 1.) T o show t h a t ~ ( x ) - - - - ~ ( y ) ~ 1, observe t h a t t h e n o r m a l bundles of ~' a n d y~ (regarded as i m b e d d i n g s of S ~ in t h e di//erentiable m a n i fold W) a r e non-trivial. T h e s e bundles are isomorphic to p : U - ~ S 5. L e t K be the THOM c o m p l e x of this bundle. (I. e., t h e space o b t a i n e d b y collapsing t h e b o u n d a r y of U to a point.) I t is well k n o w n t h a t K a d m i t s a cell d e c o m p o sition S s ~ e l~ where t h e a t t a c h i n g m a p S ~ -~ S 5 is a r e p r e s e n t a t i v e of t h e WHiTEHeAD p r o d u c t [i5,/5]. On t h e other h a n d , t h e THOM c o n s t r u c t i o n provides a m a p /o : Mo -~ K s u c h t h a t /*(e 0 ~ X , the d u a l class of ~' : Ss--+Mo, and / * ( u 2 ) [ M o ] : 1, w h e r e e 1 generates H S ( K ; Z ) a n d u 2 g e n e r a t e s H~~ A m a p / : M0 --> ~ S ~ is o b t a i n e d b y c o m p o s i t i o n of [o w i t h t h e usual inclusion S s ~ e l~ --> D S e. (Recall t h a t ~2S ~ h a s a cell d e c o m p o s i t i o n D S 0 ~-- S 5 ~ e 10 ~ e 15 - e20 . . . . , w h e r e t h e a t t a c h i n g m a p of e l~ r e p r e s e n t s [i5, i~].) Then, /: Mo---~..QS ~ has the properties /*(el)-----X, / * ( u 2 ) - - ~ l , showing t h a t ~(x) : 1. T h e s a m e c o n s t r u c t i o n applied to Y, t h e d u a l class of ~": S 5 --+ M o yields ~(y) : 1. H e n c e ~b(M0) = ~(x).q~(y) : 1. I f M ( / ~ , / 2 ) , w i t h t h e differentiable s t r u c t u r e induced b y W (of which M(/1, [~) is the b o u n d a r y ) were diffeomorphie to S ~ w i t h t h e s t a n d a r d differentiable s t r u c t u r e , t h e differentiable s t r u c t u r e on W could b e e x t e n d e d to a differentiable s t r u c t u r e o v e r t h e cone CM([~,/~), p r o v i d i n g a differentiable 19 CM'H vol. 84

266

MICHEL A. KERVAIRE

s t r u c t u r e on M o. H o w e v e r , ~ (M0) ---- 1 a n d T h e o r e m 1 show t h a t a differentiable s t r u c t u r e on M o does n o t exist. Hence, M ( / x , ~2), h o m e o m o r p h i c to S ~ is n o t diffeomorphie to S ~.

w 4. The auxiliary space Q* L e t y --_ SS v 2i5e 6 be t h e space o b t a i n e d b y a t t a c h i n g a 6-cell to S 5 b y a m a p S 6 -+ S 5 of degree 2. Lemma 4.1. Proo].

L e t o~ E ~r5 ( Y ) ~ Z2 be the generator, then

[~, ~] =fi 0 ~ =9 (Y).

W e i d e n t i f y Y with t h e STIEFEL manifold VT,~. Consider t h e e x a c t

sequence 9 ""

Since

=1o(S e) = 0,

i, [i~, i~] = [i, (i~), i,

-~

~1o(S ~) -~ ~9(S 5) J; ~9 (V7,2)

and (i~)]

[iu, is] =

[~,

~]

is non-zero in ~

-~

""

#o(SU),

it follows t h a t

o.

L e t Y* ~ Y ~, e 1~ be t h e space o b t a i n e d f r o m Y b y a t t a c h i n g a 10-cell e 1~ using a r e p r e s e n t a t i v e [ : S 9 -+ Y of [cr c,]. Since Y is 4-connected, t h e characteristic m a p

7.

(D 1~ S 9) - + ( Y * , Y)

of e 1~ induces a n i s o m o r p h i s m

A

/ , : =19(D x~ 8 9) ~ 1 0 ( Y * ,

Y) 9

( C o m p a r e J . H. C. WHITEHEAD [12], T h e o r e m 1.) T h u s t h e relative H I m E WlCZ h o m o m o r p h i s m h u : Z q o ( Y * , Y ) --~ H ~ o ( Y * , Y ) ~ Z is an i s o m o r p h i s m . Consider t h e h o m o t o p y - h o m o l o g y ladder of ( Y*, Y) : " ' " "-~ ;7/710( Y ) --> Y/:lO( Y * ) ~ Yl:lO( Y * , Y) "-~ =9 ( Y ) "+ " ' "

9..-->0

-->

H~o(Y )-+HIo(Y*,Y)--*O-->

...

9

Since a sends t h e g e n e r a t o r of glO (Y*, Y) into [c~, c~] :fi 0, a n d 2 [cr c~] ---- 0, it follows t h a t e v e r y e l e m e n t in I m (h: =lo(Y*) -+ Hxo(Y*)} can be halved. I t follows t h a t for e v e r y m a p go : S~~ -~ Y * , the induced h o m o m o r p h i s m go* : Hi~ (Y* ; Z2) -+ Hi~ ( S10 ; 7"2) is zero. L e t ~9 be t h e space of loops o v e r S 6. U p to h o m o t o p y t y p e ~9 = S 5 ~ e 1~ ~ e 15 . . . . . w i t h e 1~ a t t a c h e d b y a m a p of class [i5, ib]. L e t $9* ---- Q ~ e 6, w h e r e e e is a t t a c h e d b y a m a p of degree 2 on S s c ~9. T h e r e is a n a t u r a l inclusion Y* --~ ~ * which induces a n i s o m o r p h i s m on e o h o m o l o g y groups in d i m e n s i o n 10. Hence, we h a v e t h e Lemma4.2. Let g: SI~ ~ * be a m a p , a n d let u 2 be the generator o/ H~~ Z2) o~ Z2. T h e n , g*(u2) = O.

A Manifold which does n o t a d m i t a n y Differentiable S t r u c t u r e

267

w 5. A lemma on homotopy groups of spheres

Lemma 5.1. The map ~,+9(8 n) -+g,+~0(S=), /or n > 12, de/ined by composition with the generator ~ o/ z~+xo(S n+~) is sur]ective on the 2-component. This lemma was communicated to me without proof b y H. TODA who has also proved t h a t the 2-component of ~r~+~0(S~) is Z2. (See If. TODA [11], Corollary to Proposition 4.10.) We give a sketch of proof by computation of the POSTieIKOV decomposition modulo 2 of S n for large n, up to dimension n + 10. We begin with a remark which will yield L e m m a 5.1 whenever a long enough portion of the POSTNIKOV decomposition of S n is obtained. L e t X = K ( Z ~ , n + 9) x , K ( Z 2 , n + 10) be the space of the fibration over K ( Z ~ , n + 9) associated with the /(-invariant 1r ~ H~+ii(Z~, n + 9 ; Z2). Let / : S n+9 - , X be a map representing the generator of ~ + 9 (X) ~o Z2. Then, the composition / o V : S ~+i0 -+ X , where ~ : S ~+l~ -+ S =+~ represents the generator of nn+10(S~+9), is essential if and only if k = Sq2(e), where e is the f u n d a m e n t a l class of Hn+9(Z 2, n + 9; Z~). Since Sq2(e) generates H=+li(Z2, n + 9; Z2), it follows t h a t k =/: Sq2(e) implies k = 0. Hence, / o r is inessential if k # Sq*(e). I f k = Sq2(e), let ~'. S n+2 ~ ~en+ii -+ X ~!o ~e~+~ be the map induced by /. L e t s ~ Hn+9(S "~+~ ~,en+X~; Z2) be the generator. We identify Hn+9(X ~e~+~l; Z~) and H ~+9(X; Z~) with Hn+9(Z2, n + 9; Z~). Since /*(e) = s, and Sq~(s) =/= O, it followsthat Sq2(e) # 0 in Hn+ii(X~,en+ll; Z~). To show t h a t / o ~ is essential, it is therefore sufficient to show t h a t Sq~(e) = = 0 in H n+ll (X ; Z2). This follows from the c o m m u t a t i v i t y of the diagram 0 + - H"+9(X; Z~) ~- Hn+9(Z~, n + 9; Z2) +- 0 8 q2 ~ ,~ S q~ T

Hn+I~(X; Z2)<-- H'~+II(Z2, n + 9; Z2)+-H'~+I~

n + I0; Z~),

where the rows are t a k e n from the exact sequence of the fibration defining X (in the stable range), and ~ is the transgression. L e t Ylo ~ Yg ~ " " ---> Y~-> Y ~ - i - + " " "--)"Y o = K ( Z , n ) be the modulo 2 POSTNIKOV decomposition of S ~. (I.e., Pi : Y~ ~ Yi-1 is a fibration with fibre F~ = K(~t~, n + i), where ~ is the 2-component of the stable group ~t,+~(Sn), and H*(Y~; Z2) contains Z2 in dimension 0 a n d n, Ha (y~ ; Z2) = 0 for 0 < q < n, and H n+k ( Y~ ; Z2) = 0 for 0 < k < i + 2.) B y the ~ - t h e o r y with E ---- the class of finite groups whose order is prime to

268

MICHEL A . KERVAIRE

2, a map S ~ -+ Y, inducing an isomorphism H n(Y, ; Z~) ~ H "(S ~ ; Zz) induces an isomorphism of the 2-component of ~.+~(S ~) with ~+~(Y,) for k ~i. (Compare J . P . SERRE[8].) We have ~ a ~ Z z ~ - Z~Jr Z~ and gl0 ~ Z~ as will be seen below, thus

F s - ~ K(Zz, n + 9) • K(Zz, n + 9) • K ( Z ~ , n - ~ 9), and Lemma 5.1 follows by showing t h a t the restriction of the fibration Ylo-~Y~ over one of the factors of F 9 is K ( Z ~ , n ~ 9 ) • k K ( Z ~ , n - ~ 10) with k ~ Sq ~. This is equivalent to showing t h a t H~+~(]19; Z2)_--__Z2 is generated by a class u 9 such t h a t i*(Ug) -~ Sq2(eg), where e9 is one of the fundamental classes of Hg(Fg; Z~), and i g : F 9 ~ Y9 is the inclusion. I n a similar way, it can be read off from the tables below t h a t composition with ~ provides in]ective maps ~+~(S ~) | Zz-->~.+s(S ") and g~+s(Sn) --~ ~.+9(S ") in the stable range. Using ~ ( $ 0 ~ ) ~ Z, gs(S0~) ~ Z~, and ~ ( S O , ) ~ Z~, this implies t h a t J9:u~(SO,) -->~+9(S n) is a monomorphism. We proceed to a partial description of the modulo 2 cohomology of the spaces Y~. H* (Yo) is given by J. P. S~.RR~ in [9]. This result of J. P. S~RRE and the AD~.M relations between the STE~.NROD squares are the essential tools in computing H* (Y~ ; Z~) for /~ :> 0. Since we stay in the stable range, the spectral sequences of p~ : Y~ --~ Y~_l reduce to exact sequences T

tk

"'" <-- Hn+q+l(Y~-I) <-- Hn+q (Fk) <-- Hn+q (Y~) ~-- Hn+q (Yk-1) <-- 9 "" 9 I t is therefore sufficient to determine at each step the kernel and the image of the transgression 3. Since the cohomology of Yk is independent of k up to dimension n, we omit to mention the non-vanishing cohomology groups in dimension ~ n. The direct sum of the subgroups of H* (Y~ ; Z~) in dimensions > n is denoted H +(Yk). The symbol qk stands for the composition Pl ~ P~ o 9 9 9 o Pk, and e~ denotes the fundamental class of Hn+k(G, n ~-/c ; G). I omit Y1 and Y~ whose cohomology is straightforward, but has to be computed up to dimension n ~ 17 and n ~ 16 respectively. Hn+4(Y2; Z~) is generated by q*(Sq4eo), and Hn+5(Y~; Z2) by a class u~ such t h a t

Fa -~ K(Zs, n + 3), with ~(e~) = q*(Sq4so) and ~(flea) = u2, where fl is the BOCKST~r~ operator associated with the sequence of coefficients 0 -~ Z~ -~ Z1, -~ Z8 -~ 0, and s~ is the rood. 2 reduction of es.

A Manifold

which

does not admit

any Differentiable

269

Structure

H+ ( Ya) has a basis consisting o~ us in dimension n ~- 7, such t h a t i*(u3) : Sq~ea;' Sq~(u3), q*(SqSto) ; Sq~(u3), v3 such t h a t i*(v3) : SqUflta; Sqa(u3) ; Sqa(ua) ; Sqh(u3), Sq'Sql(ua), q*a(Sq~2to) ; Sq~(u3), Sq4Sq2(u3), ~gq~(v3) ;

S q ' Sq~(u.~) , Sq~ Sq2(u3) , q* (Sq~to) ; SqS(u3), SqTSq~(u3), SqeSq~(u3), Sq~(v~), q*(Sq~eo) ; . . .

Y4= Ya:

Y3. ( z ~ 4 : n ~ = O . )

F. : K(Z~, n ~-

6) with ~(e6) : p*p*(u3). H+ ( Ye) has a basis consisting o/ ue such t h a t i*(u6) : S q ~ S q l t 6 , q*(SqSto). , PsPhP4(%), * 9 . Sql(uB) ; nothing in dimension n -~ 11 ; q*(SqUto), Sq2Sql(u6) ; P69 Ph* P4* (Sq 4 va), Sq~(u6) , v6 such t h a t i* (v6) : SqTt6, 9 q*(SqX, to), Sqh(ue);q*(Sq~eo), p~* p~* p,* (Sq e v3), 9 9 9 (and possibly o t h e r classes of dimension n ~- 15).

F 7 = K(Z16 , n - ~ 7) with v(t~7) = q*6(SqSto) a n d v(fl'tT) : P,*Pa*P4(Va), where fl' is t h e BocKsTEI~ o p e r a t o r of 0 --> Z~ --> Z3~ -~ Z16 --> 0, a n d e~ is t h e r e d u c t i o n m o d u l o 2 of fT.

H+ ( YT) has a basis consisting o/ u , in dimension n + 9, such t h a t i*(uT) : Sq~(e1~), p*(u6); Sql(uT), p*(Sqlus), v7 such t h a t i*(vT) : SqZfl'e7;

Sq 1(v7) ; Sq"Sq 1(u7), p* (Sq~Sq lu.) . . . .

(Sq"(vT) = 0.)

F s:K(Z~+ Z2, n + 8) w i t h ~ ( t ~ ) : u T , z ( t ~ ) : p * ( u s ) , u e8 a r e the t w o f u n d a m e n t a l classes in H n + 8 ( F s ; Z2).

where ~s' a n d

H+ ( Ys) has a basis consisting o/ p*(vT), us, vs, where % (Us) = Sq2(t~) a n d i*(vs) : Sq~(t~);

Sql(us), Sq~(vs), p*(Sq~v,) ; Sq2(us), Sq~(vs) . . . . l t 9rl w h i c h F 9 ~ K(Z~ + Z2 + Z2, n + 9) w i t h f u n d a m e n t a l classes to, to, are send b y transgression on p*8(vT), u s, v s respectively. Hn+Xz(Yg; Z2) ~ Z2(uo) ,

where

i9*(q~9) :

Sq~(eg)

.

W e h a v e seen t h a t this s t a t e m e n t implies L e m m a 5.1, hence t h e p r o o f is complete.

Institute o/Mathematical Sciences, New York University

270

MICHELA. KERVAIRE A Manifold which does not admit any Differentiablo Structure BIBLIOGRAPHY

[1] R. Bol~r and H. SAMELSON,On the Pontryagin product in ~paces o] paths. Comment. Math. Helv. 27 (1953), 320-337. [2] S. T. Hu, ttomotopy theory. Academic Press, 1959. [3] M. A. KERVAXRE, Relative characteristic classes. Amer. J. Math., 79 (1957), 517-558. [4] M. A. KERV~RE, Some non-stable homotopy groups of LIE groups. Illinois J. Math., 4 (1960), 161-169. [5] J. MrLNOR, Di]]erentiable structures on spheres. Amer. J. Math., 81 (1959), 962-972. [6] J. MI~NOR, A procedure ]or killing homotopy groups o] di]]erentiable mani]olds. Proceedings of the Symposium on Differential Geometry. Tucson, 1960, to appear. [7] J. M~NOl~ a n d M. A. KERVArRE, BERNOULLI numbers, homotopy groups, and a theorem o/ RennIN. Proceedings of t h e Int. Congress of Math., Edinburgh, 1958. [8] J. P. SERRE, Groupes d'homototde et classes de groupes abelicns. Ann. of Math., 56 (1953), 258-294. [9] J. P. SERm~, Cohomologie modulo 2 des complexes d' EILE~ERG-MAcLA~. Comment. Math. Helv., 27 (1953), 198-232. [10] R. T~oM, Quelques propri~t~ globales des vari~t~s di]]~rentiables. Comment. Math. Helv., 28 (1954), 17-86. [11] H. TODA, On exact sequences in STEENROD algebra mod. 2. Memoirs of the College of Science, University of Kyoto, 31 (1958), 33-64. [12] J. H. C. W~ITEHEAD, Note on suspension. Quart. J. Math. Oxford, Ser. (2), 1 (1950), 9-22.

Received May 7, 1960