On deforming G-maps to be fixed point free (Q1088214)

Let G be a compact Lie group, M a compact, smooth G-manifold and \(f: M\to M\) a G-map. If for each isotropy subgroup H with finite Weyl group WH, \(M^ H\) is simply connected of dimension \(\geq 3\) then f is G-homotopic to a fixed point free map if, and only if, the Lefschetz number \(L(f^ H)=0\) for each such H, where \(f^ H=f| M^ H\). This result is due to \textit{D. Wilczyński} [Fundam. Math. 123, 47-60 (1984; Zbl 0548.55002)]. We prove, without assuming simple connectivity on \(M^ H\), that f is G-homotopic to be fixed point free if, and only if, the Nielsen number \(n(f^ H)=0\) for all H with finite WH. There is also a codimension condition.