Wu numbers of singular spaces
This interesting paper deals with the motivational questions behind the development of intersection homology: ``For how singular a space is it possible to define cobordism invariant characteristic numbers?'' There are many singular spaces for which Whitney, Chern, and L classes can be defined; however, these are homology classes which cannot necessarily be multiplied to give characteristic numbers. \textit{M. Goresky} and \textit{R. MacPherson} [Topology 19, 135-165 (1980; Zbl 0448.55004)] hoped that these classes could be lifted to intersection homology groups where their products could be formed. Then, as a space becomes more singular it would be possible to multiply fewer characteristic classes and the corresponding cobordism groups would be determined by fewer characteristic numbers. This program has so far failed except in extreme cases: either where there is a single characteristic number as in Sullivan's theory of mod 2 Euler spaces, or in P. Siegel's theory of mod 2 Witt spaces where the intersection homology Euler characteristic is the only cobordism invariant. The paper under review exhibits four new classes of singular spaces, with increasingly severe singularities for which various cobordism invariant characteristic numbers can be constructed, and for which these characteristic numbers completely determine the cobordism groups. These numbers are obtained by lifting Wu classes, rather than Stiefel-Whitney classes, to intersection homology and multiplying them. These four classes are: orientable \(\bar s-\)duality spaces, orientable locally square-free spaces, locally orientable Witt spaces, and locally orientable spaces. Each class requires different geometrical techniques, but they all use surgery on singular spaces (which is surprisingly simpler than surgery on manifolds) and they involve the understanding of the geometrical consequences of the vanishing of a Wu number.
cobordism invariant characteristic numbers
scientific article; zbMATH DE number 4140838
scientific article