Poincar� spaces, their normal fibrations and surgery

Abstract

Let X be a Poincare space of dimension rn, i. e. there is an element [X]EHm(X) such that [X]n: Hq(X)--+Hm_q(X) is an isomorphism for all q. An oriented spherical fibre space ~, Eo ---.!4 X with fibre F homotopy equivalent to Sk-\ is called a Spivak normal fibre space for X if there is IXE7rm+k(T(~)) of degree 1, i.e. such that h(lX)n U~=[X], where T(~)=X un ('(Eo), U~E Hk(T(~)) is the Thorn class; h: 7r* --+ H* the Hurewicz homomorphism. Let us always assume that X is the homotopy type of a CW complex. Differential topology and surgery theory in particular has had great success in studying the problem of finding manifolds of the same homotopy type as a given Poincare space, and classifying them (see [2] for example). In this theory, one finds weaker structures for Poincare spaces analogous to well known structures on manifolds, and then studies the obstructions to lifting the structure to the strong type found on a manifold. For example, the Spivak normal fibre space is the analog of the stable normal bundle to the embedding of a manifold in the sphere, and the problem of making this spherical fibre space into a linear bundle (up to fibre homotopy type) is central to the theory of surgery. In his thesis Spivak [5] showed that if 7rl X =0 then a Poincare space X has a Spivak normal fibre space for k>dim X, unique up to fibre homotopy equivalence (see also [2, Ch. I, § 4]). His proof extends to nonsimply connected Poincare spaces X, provided that Poincare duality holds with local coefficients and X is dominated by a finite complex [6J. However, the proof of uniqueness, due essentially to Atiyah [1] makes use of only ordinary homology, and Spanier-Whitehead S-theory. Hence it is quite plausible to think that the Spivak normal fibre space exists also, without any assumptions but Poincare duality with ordinary integer coefficients. We shall show that this is in.fact the case.