Pulling back fixed points

Abstract

If X, Y are G-spaces (spaces on which a compact Lie group G acts), and f: X ~ Y is a G-map, then the image of a fixed point in X is fixed in I1. When does the inverse image of a fixed point contain a fixed point, in particular if X, Y are G-manifolds and f has degree prime to ]GI (=order of G). In (unpublished work of the author1), this is shown for elementary abelian p-groups, with X, Y more general spaces ("n-near manifolds"). In this paper we show that this is true for abelian p-groups G acting smoothly on X, with p odd or with p=2 with an additional complex hypothesis. We give a counterexample for nonabelian G. The method is to prove a generalization of a theorem of (Bredon, 1973) for 7lip actions, which says that maps of degree prime to p induce injections on the mod p cohomology of the fixed set. We generalize this theorem to abelian p groups, where the action on X is smooth, and p 4= 2 (or other hypothesis). This seems to be new even in the case of elementary abelian p-groups of rank > 1. We actually prove a stronger theorem for a generalization of the notion of degree, which may be non-zero for maps between manifolds of different dimension. The proof involves complex linear equivariant K-theory in an essential way, which requires smoothness and p + 2, or a complexity hypothesis on X in order to produce the appropriate complex linear bundles. Among the corollaries, we show that an abelian p-group (p odd) cannot act smoothly on a closed manifold with exactly one isolated fixed point, generalizing an old result of (Atiyah and Bott, 1964) for G=Z/p. This was found independently by (Ewing and Stong, 1986) by a different argument. I am indebted to Ib Madsen for his comments, in particular for discussions on equivariant K-theory. Thanks are due also to Michael Davis for carefully reading the manuscript, and pointing out several difficulties. I also wish to thank the referee for his comments.