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Betti numbers of fixed point sets and multiplicities of indecomposable summands. (English) Zbl 1031.55003

Let \(G\) be a finite group. By a \(G\)-Moore space, we mean a simply-connected \(G\)-space \(X\) such that \(H^i(X,x_0;k)=0\) except for \(i=n\) for some fixed \(n\geq 2\). Suppose \(G\) has even order, \(k\) is a field of characteristic \(2\), and \(M\) is a \(kG\)-module with \(H^n(X,x_0;k)=M\) for some \(G\)-Moore space \(X\). The main objective of the paper under review is to obtain an explicit formula for the total Betti number \(\beta(X^C)\) of the fixed point set \(X^C\) when \(C\cong \mathbb Z_2\) or \(\mathbb Z_4\), the cyclic group of order 2 or of order 4. The main technique is an isomorphism theorem on \(C\)-equivariant cohomology via localization.

MSC:

55M35 Finite groups of transformations in algebraic topology (including Smith theory)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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