The Lefschetz number for equivariant maps
Suppose G is a compact Lie group and X is a G-ENR, that is, a G-space which is a G-retract of a G-invariant open subset of a Euclidean G-space. Let \(f: X\to X\) be a G-map and let f G: X \(G\to X\) G be its restriction to the G-fixed point set of X. Denoting Lefschetz numbers by \(\lambda\), the main results are: (a) if the G-action on X is semifree, then \(\lambda\) (f) and \(\lambda\) (f G) are congruent modulo the Euler characteristic of G; (b) if G is finite of order p k, where p is a prime, then \(\lambda (f)=\lambda (f\) G) mod p; (c) if G is a torus, then \(\lambda (f)=\lambda (f\) G). It then follows, for instance, that if G if of order p k and \(\lambda\) (f) is not divisible by p, then f has a fixed point in X G. Note that there are no freeness assumptions on the G-action in (b) and (c). All these results, and some others, follow from a single argument based on techniques the author developed previously [Osaka J. Math. 20, 553-572 (1983; Zbl 0523.57025)] that are used to G-homotope f to a G-map h with a very well-behaved fixed point structure. Then the fixed point index for (nonequivariant) maps on ENRs [\textit{A. Dold}, Topology 4, 1-8 (1965; Zbl 0135.231) or the reviewer, ``The Lefschetz fixed point theorem'' (1971; Zbl 0216.196)] is used to relate indices of h and h G under the various hypotheses.
G-retract of a G-invariant open subset of a Euclidean G-space
scientific article; zbMATH DE number 4046687