Spin manifolds with nonzero dual Stiefel-Whitney classes of large grading

Abstract

The dual Stiefel-Whitney classes wbar_j(M) of a manifold M are elements of H^j(M;Z/2) which give information about embedding and immersing M in Euclidean space. We consider the question of finding, for each n, the largest j such that there is an n-dimensional Spin manifold with wbar_j nonzero. We obtain upper and lower bounds. Since one theorem proves existence of manifolds without giving explicit examples, we consider a parallel question of finding explicit manifolds with wbar_j nonzero for j as large as possible.