Equivariant Lefschetz classes in Alexander-Spanier cohomology (Q1356401)

Let \(G\) be a finite group, \(X\) a paracompact Hausdorff \(G\)-space and \(f:X\to X\) a \(G\)-map. If \(X^H\) is the fixed point subspace for a subgroup \(H\leq G\), then \(f\) induces a map \(f^H: X^H\to X^H\). Under the assumption that the Alexander-Spanier cohomology \(H(X^H,\mathbb{Z})\) is finite for every \(H\leq G\) it is possible to define the Lefschetz number of \(f^H\), \(L(f^H)\in \mathbb{Z}\). Then the equivariant Lefschetz class of \(f\) is \(L(f)= (L(f^H))\in \sqcap_{\psi(G)} \mathbb{Z}\), where \(\psi(G)\) is the set of conjugacy classes of all subgroups of \(G\). The authors consider a suitable additional condition on the equivariant Alexander-Spanier cohomology of \(X\) to prove that \(L(f)\in A(G)\), where \(A(G)\) is the Burnside ring of \(G\). The cases when \(G\) is a compact Lie group and when \(G\) acts on finite contractible complexes are also investigated. The authors classify fixed point sets of such an action.