Abstract
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 orderqacts semifreely on a sphere, the fixed set is a modqhomology sphere. Conversely, given a modqhomology 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 Jones and then by many others. In this note, we give new numerical congruences satisfied by the homology of the fixed sets and give a definitive solution to the problem for characteristic fixed-point sets. c !1999 John Wiley & Sons, Inc.
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