Wu numbers of singular spaces

Abstract

IN THIS paper we consider the following question: for how singular a space is it possible to define cobordism invariant characteristic numbers? This question was one of the motivations behind the development of intersection homology: there are many singular spaces (complex algebraic varieties, for example) for which Whitney [40], Chern [28], and L [ 1 S] classes can be defined, but these are homology classes and cannot necessarily be multiplied so as to give characteristic numbers. It was hoped ([18]) that these classes could, in certain cases, be lifted canonically to intersection homology groups where their products could be formed. Thus, as a space is allowed to become more and more singular, it should become possible to multiply fewer and fewer characteristic classes, and so the corresponding cobordism groups would be determined by fewer and fewer characteristic numbers. This approach has (so far) failed completely, except in the “extreme” cases where there is a single characteristic number (for example, in Sullivan’s theory of mod 2 Euler spaces ([40], Cl]), where the Euler characteristic is the only cobordism invariant, or in P. Siegel’s theory of mod 2 Witt spaces ([37], [ 15]), where the intersection homology Euler characteristic is the only cobordism invariant. We will exhibit four interesting classes of singular spaces with increasingly severe singularities: