One fixed point actions on spheres, I

Characteristic Fixed-Point Sets of Semifree Actions on Spheres

A group action is semifree if it is free away from its fixed-point set. P. A. Smith showed that when a finite group of order q acts semifreely on a sphere, the fixed set is a mod q homology sphere. Conversely, given a mod q homology sphere as a subset of a sphere, one may try to construct a group action on the sphere fixing the subset. The converse question was first systematically studied by J...

Fixed Point Sets of Involutions on Spheres

Group Actions on Spheres with Rank One Isotropy

Let G be a rank two finite group, and let H denote the family of all rank one p-subgroups of G, for which rankp(G) = 2. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X ' S with isotropy in H, whose fixed sets are homotopy spheres.

Group Actions on Spheres with Rank One Prime Power Isotropy

We show that a rank two finite group G admits a finite G-CW-complex X ' S with rank one prime power isotropy if and only if G does not p′-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G.

Fixed Point Formulas and Loop Group Actions

In this paper we present a new fixed point formula associated with loop group actions on infinite dimensional manifolds. This formula provides information for certain infinite dimensional situations similarly as the well known Atiyah-BottSegal-Singer’s formula does in finite dimension. A generalization of the latter to orbifolds will be used as an intermediate step. There exist extensive litera...