The first exotic class of a manifold

Abstract

Let BF be the classifying space for stable oriented spherical fibrations. Gitler and Stasheff have defined a cohomology class el in H pr -l(BF; Zp), where here (and throughout this paper) p is an odd prime and r=2(p1). For a discussion of the nature and significance of this class, see the introductions to [2], [4], [7]. Suppose P is a (pr1)-dimensional oriented Poincare complex. Let v be the stable normal spherical fibration of P, i.e., the unique stable spherical fibration with reducible Thom complex [6]. Define e1(P) to be el(v). Similarly, define qi(P) to be qt(v), where qi is the ith Wu class. It is clear that e1(P) depends only on the homotopy type of P. We wish, however, to express e1(P) in terms of an explicit invariant of the homotopy type of P. In fact, we will construct a certain nonstable secondary cohomology operation Q, mapping cohomology classes of dimension r into classes of dimension pr-1 (Zp coefficients), such that