Relations among characteristic classes

Abstract

Let M be an n-dimensional, compact, closed, ${C^\infty }$ manifold, and $v:M \to BO$ be the map classifying its stable normal bundle. Let $S \subseteq {H^\ast }(BO;{Z_2})$ be a set of characteristic classes and let q, k, be fixed nonnegative integers. We define $I_n^q(S,k) = \{ x \in {H^q}(B):{v^\ast }(x) \cdot y = 0$ for all $y \in {H^k}(M;{Z_2})$ and for all n-dimensional, ${C^\infty }$ closed compact manifolds M, which have the propery that ${v^\ast }(S) = \{ 0\} \}$. In this paper we compute $I_n^q(S,k)$, where all classes of S have dimension greater than $n/2$. We examine also the case of BSO and BU manifolds.