Wu numbers of singular spaces
type: Item
instance of: scholarly article
title: Wu numbers of singular spaces
zbMATH Open document ID: 0696.57011
DOI: 10.1016/0040-9383(89)90012-8
author: M. Goresky
author: William Pardon
published in: Topology
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4140838
zbMATH Keywords: intersection homology
zbMATH Keywords: cobordism invariant characteristic numbers
zbMATH Keywords: cobordism groups
zbMATH Keywords: Euler spaces
zbMATH Keywords: Witt spaces
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Wu classes
zbMATH Keywords: duality spaces
zbMATH Keywords: orientable locally square-free spaces
zbMATH Keywords: locally orientable Witt spaces
zbMATH Keywords: locally orientable spaces
zbMATH Keywords: surgery
reviewed by: R. J. Stern
MaRDI profile type: Publication
full work available at URL: 0040-9383(89)90012-8
OpenAlex ID: W2045095153
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description: scientific article; zbMATH DE number 4140838