When f : M → M is a
self-map of a compact manifold and dimM ≥ 3, a classical theorem of Wecken states
that f is homotopic to a fixed point free map if, and only if, the Nielsen number n(f)
of f is zero. When M is simply connected, and dimM ≥ 3 the NASC becomes
L(f) = 0, where L(f) is the Lefschetz number of f. An equivariant version of the
latter result for G-maps f : M → M, where M is a compact G-manifold, is due to D.
Wilczyński, under the assumption that MH is simply connected of dimension
≥ 3 for any isotropy subgroup H with finite Weyl group WH. Under these
assumptions, f is G-homotopic to a fixed point free map if, and only if, L(fH) = 0
for any isotropy subgroup H (WH finite), where fH= f|MH and MH
represents those elements of M fixed by H. A special case of this result was
also obtained independently by A. Vidal via equivariant obstruction theory.
In this note we prove the analogous equivariant result without assuming
that the MH are simply connected, assuming that n(fH) = 0, for all H
with WH finite. There is also a codimension condition. Here is the main
result.
