Intersection homology and free group actions on Witt spaces
type: Item
instance of: scholarly article
title: Intersection homology and free group actions on Witt spaces
zbMATH Open document ID: 0755.57017
author: Stephen J. Curran
published in: Michigan Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Witt spaces, \(X^ n\), are \(n\)-dimensional pseudo-manifolds verifying a link condition; they satisfy a Poincaré duality theorem using the intersection homology \(IH_ *^{\overline m}(X,\mathbb{Q})\) [\textit{M. Goresky} and \textit{R. MacPherson}, Topology 19, 135-165 (1980; Zbl 0448.55004)]. In the paper under review, the author studies free \(G\)- actions of a finite group \(G\), on compact Witt spaces. First, he obtains results on Euler characteristic of \(X^ n\), \(\chi(X)=\sum^ n_{i=0}(-1)^ i\dim IH_ i^{\overline m}(X;\mathbb{Q})\), and Euler semicharacteristic of \(X^{2n+1}\), \(\chi_{1/2}(X)=\sum^ n_{i=0}(-1)^ i\dim IH_ i^{\overline m}(X;\mathbb{Q})\): 1) If \(X^ n\) has a free \(G\)-action that acts trivially on the intersection homology, then \(\chi(X)=0\). 2) With the same hypothesis on \(X^{4n+1}\), then \(\chi_{1/2}(X)\) is even or \(G\) is a direct product of a cyclic 2-group and an odd order group. Next, the author considers the \(L\) class, \(L(X)\), defined by Goresky and MacPherson, of \(X^{4n}\). Suppose \(X^{4n}\) has a free \(G\)-action that acts trivially on the fundamental group, \(\pi\), and trivially on the intersection homology of \(X\) with any local coefficient system. Consider \(p:\pi\to\text{Sp}(2l,\mathbb{R})\) a representation of \(\pi\) into the symplectic group and \(f: X\to B\pi\) the classifying map of the universal cover of \(X\). Then, for all \(y\in H^*(B\text{Sp}(2l,\mathbb{R});\mathbb{Q})\), one has \(\langle (Bp)^*u,f_ *L(X)\rangle = 0\).
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 63016
zbMATH Keywords: Witt spaces
zbMATH Keywords: \(n\)-dimensional pseudo-manifolds
zbMATH Keywords: link condition
zbMATH Keywords: intersection homology
zbMATH Keywords: free \(G\)-actions
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Euler semicharacteristic
reviewed by: Daniel Tanré
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004459
OpenAlex ID: W2035409594
DOI: 10.1307/MMJ/1029004459
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description: scientific article; zbMATH DE number 63016