Intersection homology and free group actions on Witt spaces
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 \(\langle (Bp)^*u,f_ *L(X)\rangle = 0\).
scientific article; zbMATH DE number 63016