In this note we construct a family of odd-dimensional, closed, combinatorial manifolds, none of which has the homotopy type of a closed differentiable manifold. These manifolds all have an infinite cyclic fundamental group. W. Browder [1] and S. P. Novikov [9] have proved that a simply-connected finite complex, K, satisfying Poincaré duality with respect to an odd-dimensional fundamental class ue H2n+X(K), («2:2) has the homotopy type of a closed (2« + l)dimensional differential manifold provided there exists a vector bundle, f, over K whose Thorn space T(Ç) has a spherical top homology class (i.e., TTqT(£-)» HqT(^) is surjective for q = 2« +1 + dim £). Denoting one of our combinatorial manifolds by M, we prove that SM, the suspension of M, has a spherical top homology class, so that the trivial real line bundle over M satisfies the Browder-Novikov hypothesis. The manifolds, M, show that the Browder-Novikov theorem cannot be extended to the nonsimplyconnected case, at least not without additional hypotheses on K. (Recall that the Thorn space of the trivial real line bundle over a space X has the homotopy type of S1 y SX.) The manifolds M2n +1 are constructed from certain knotted homotopy (2n- 1)spheres in S2n +1 (n is odd). Let A be (the space of) the tangent unit disk bundle over Sn, and W the differential manifold with boundary obtained by plumbing together two copies Ax, A2 of A. (See [4], [5], or [8].) The images Sx, S2 in W of the zero cross-sections in Ax and A2 respectively have a single (transversal) intersection point. Denote by 22""1 the boundary of W. It was proved in [8], or more generally follows from Smale theory, that £2n_1 is combinatorially equivalent to S2"-'1. We now imbed W into S2n +1. It is well known, and easy to see, that W can be differentiably imbedded into S2n +1 so that if v denotes a normal vector-field on W in 5'2n+1, and if S'x, S'2 are the translates of Sx, S2 by a small positive amount s along v, then \=L(Si, 5¡) for ¡'=1, 2, are odd integers, where L( , ) denotes the linking coefficient in S2""1"1(.For further remarks on this see the lemma on normal bundles at the end of this paper.)

+1 given by </>° (nxid) is a combinatorial imbedding. Let Af2n+1= y(S2n+1, fa) be the combinatorial manifold obtained from S2n +1 by spherical modification:

M2n +i = (^«"-^»-'xS^u^xJ1, where F2 = int D2, and (x, j) e S2"-1 x S1 <=T)2"x S1 is identified with fax, y). As usual we denote by fa: D2nx S1 -> M the imbedding induced by the inclusion.

Proposition. Fne manifolds M2n +1 constructed above for odd n have the following
properties for n > 1:
(

(

Remark. Since isomorphic Z[7]-modules have identical elementary ideals, and
the 0th elementary ideal of the module under (

The proofs of (

M-fa^xS1)

= S2»"-^2»-^/)2).

Hence irlM=TTi{S2n+1?bW) for i<2n-1.

I:TTx{S2n+1-bW)^Z is given by assigning to a etrx{S27l+1-bW) the intersection coefficient with W of a representative curve/: S1 -> S2n+1-bW. Clearly, 7 is surjective. To prove that 7 is injective, let U be a tubular neighborhood of W and let Y=S2n+1-U. Then first, 71-^=0} by the van Kampen theorem, since W and hence b Y are simply connected. Secondly, ^(Y, bY)={l} by the homotopy exact sequence of {Y, bY). An element ete Ker 7 can be represented by a differentiably imbedded curve [0, 1] -*?S2n+1-bW which intersects W transversally in a finite number of interior points. Since 7(a) = 0, there will be a pair of consecutive intersection points with IKhaving oppositive intersection coefficients. The arc joining these points represents an element of ttx( Y, b Y) and thus is homotopic to an arc in U which can then be pushed away from W, reducing the number of intersection points by 2. Eventually, we get a representative of the given element a whose image is contained in Y. Since Y is simply connected, a=l.

To calculate 7r¡Af for i^n, denote by W+ and W_ the two copies of W in bU=bY. Let Xx = S2n +1?bW. Following Seifert, we construct the universal covering X of Xx as the union of countably many copies of Y which we denote by Yk, keZ, with W%identified with W1+1 for every k. Then ?rri(S2n+1-bW) = 0for 2¿/'<« follows immediately from 77^=0 for /'<«. The module ttn(S2n+l ?bW) is isomorphic to Hn(X) for which we get a presentation by the Mayer-Vietoris theorem. If we denote by ¿¡x,f2 the generators of HnW represented by Sx, S2 respectively, v+ and v_ the obvious mappings of W onto W+ and W-, then a presentation for Hn(X) is obtained from a presentation of 77?F (as an abelian group) by adjoining the relations ft»+(&)=*_ (ft). Now, it is easy to see that HnY is free abelian on 2 generators xx, x2, and if S is a sphere in Y, the class of 5 is given by (S) = L(S, Sx)xx+L(S, S2)x2.

The presentation for Hn(X)^TrnM claimed in (

The proof of (

To prove (

K = (S1 V SI V ???V S;)uÉ>ï +1u---u<?2+1ue2nue2n+1. (In fact, it can be proved using Smale theory that M has a handle decomposition inducing the above cell structure. See [4].) Since HnM = 0, we have, up to homotopy type

SM = (>S'2u/e2n + 1)u!,e2n +2.

Now, since H1M=Z, there is a map M -* S1 whose composition with the inclusion S1 ->- M is homotopic to the identity S1 -^ S1. Taking the suspension of these maps we see that S2 is a retract of M. It follows that the attaching map/is trivial, and thus

SM = (S2 v S2n + 1)\j9e2n +2, up to homotopy type. Since H2n+2M=Z, it follows that g must have degree 0 (on S2n+1), and since Tr2n+ 1(S2vS2n+1) = TT2n+x(S2) + TT2n+ x(S2n + 1), g is homotopic to a mapping « into S2. Thus S M has the homotopy type of

S2n+1 y (£2 Ugft e2n + 2)_ Using again the retraction SM ->?S2, we see that h is trivial. So SM has the homotopy type of

S2 v S2n+1 v S'2n+2.
(

The proof of (

D2xS2n-\
we have an imbedding/: 52n_1 -> 5'2n+1. (It is however essential that S2"'1 here
be the sphere with the usual differential structure.) An argument similar to the one
used in the proof of (

{t-l)A (r-l)F'tE {t-l)B+E {t-l)C where F is the sxs identity matrix, B' is the transpose of F and A = ||F(|,+ , fj)||, B= ¡F(£+, r¡})\\and C= \\Ctj\\= \\L{r¡t+r,¡,)\\. (We remark that the matrices A and C are symmetric (because n is odd).) Using the lemma below we recognize an mod 2 as the obstruction to trivializing the normal bundle in V to an imbedded sphere representing $¡. A similar relation holds between cu and ^ and so c(^)= s 2 a'iCumoc*^ i=i is the Arf invariant of the quadratic form of V as defined in [6].

Now, det R is the Oth elementary ideal of Trn(M) = TTn(S2+n x?/(52n_1))and hence det R and (A^ ?A2+ A)(i?l)2 + r must generate the same ideal in Z[J]. An argument of Robertello's [10], sketched below for the convenience of the reader, shows that detR = ts-1(c(V)(t2+l) + t) modulo the ideal generated by 2 and (r-1)4. Since (A1A2-A2+ A)(r-l)2 + i = (r2 + l) + rmod2, it follows that C(F) = lmod2 but since SFis diffeomorphic to S2"-1 this contradicts the Brown-Peterson result [2] when « = 4fc+l, Aral.

Robertello's argument in brief is this. Let R = (x¡,)?thus xi} is divisible by t? 1 except if l^i^s and j=i+s or 1 újús and i=j+s. Let Sa¡Bbe the set of permutations (ix, ...,/'2s) for which ik^s + k for exactly a values of ke[l,s] and is+k^k for exactly ß values of k e [1, s]. Thus det 7? = 2,

2, xi.iix2.i2- ? ??fas.ia.

Now the individual terms in saS for a, ß e [2, s] are divisible by (t? l)4 so that we need only consider the first few S^'s. 50j0 contains only the permutation (s+l, ...,2s,l,2,...,s?l,s) which gives rise to the term

Il (=i

[(t-t)bHt]l(t-l)bH+l] = rsmod2.

2Í n i.k U'VJ*t

n J U = l;i*k ((t-i)bjj+i)\(t-i)2aikCkl

J which mod {2, (r-1)4} is ts~1(l + t2)c(V). Further similar calculation shows that 2s2 ! +2si 2 =0- (Essential use is made of the fact that A and C are symmetric.)

Before stating the lemma on normal bundles, recall that an «-dimensional bundle, £\ over an «-sphere determines an element [Çn]e irn (BSOn)?where 7?SOnis a classifying space for the group 50«. We denote by Tn the tangent bundle of Sn.

Lemma on normal bundles. Let Sn<=S2n+1 be a differentiable imbedding, v a never vanishing normal field, Sn<^S2ri +1 a disjointly imbedded sphere obtained by "pushing" Sn along v and finally r¡, the complementary normal bundle?i.e., -n(x) = the vectors normal to Sn at x but perpendicular to v(x). Then

[v]=L(S",S")[Tn]eTTn(BSOn).

Proof. There is no loss in generality (see [3]) in assuming that the imbedding is the usual one?to wit Sn<=Snx RcR" +i = R* +ixO<=Rn + 1 x Rn = S2n +1-oo. Thus we may refer the normal vector field, v, to the standard framing of this normal bundle?thus v and r¡ are described completely by a function/: Sn-> Rn+1 ?0. Since the entities involved in our assertion are unchanged if we vary v (through never-zero normal fields) we may assume that /» is a differentiable map to Sn having the south pole as a regular value. Then it is clear that 5" intersects Dn +1 x 0 transversally once for each inverse image of the south pole and in fact L{Sn, Sn) = the (algebraic) number of such inverse images = the degree of/. On the other hand, we clearly now have t)={{x, v) e Snx Rn +1 |/v(x)J_¡;} and this is obviously the "pull-back" under/ of the tangent bundle of Sn. Thus M = deg/[Fn].