A Möbius inversion formula for generalized Lefschetz numbers (Q1412387)

Let \(G\) be a finite group, \(X\) a \(G\)-ENR and \(f\colon U\to X\) a compactly fixed \(G\)-map defined on an open subset \(U\) of \(X\). For any isotropy group \(H\) of \(G\), the restriction of \(f\) to the subspace fixed by \(H\) is denoted by \(f^H\colon U^H\to X^H\). Under some assumptions on \(f\), the author obtains some relations between the generalized Lefschetz numbers \(L(f^H| U_H)\) and \(L(f^K)\) for all isotropy groups \(K\supset H\). Moreover, the author proves a converse of the Lefschetz property, namely, the vanishing of generalized Lefschetz numbers implies, in some sense, that the given \(G\)-map may be \(G\)-homotopic to a fixed point free map. Some interesting examples are presented.