Simple-Connectivity and the Browder-Novikov Theorem
M. A. Kervaire, A. Vasquez · Transactions of the American Mathematical Society · 1967
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.)