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Characteristic fixed‐point sets of semifree actions on spheres
Author(s) -
Davis James F.,
Weinberger Shmuel
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199908)52:8<935::aid-cpa2>3.0.co;2-t
Subject(s) - mathematics , fixed point , converse , homology (biology) , congruence relation , spheres , combinatorics , group (periodic table) , pure mathematics , discrete mathematics , mathematical analysis , geometry , biochemistry , chemistry , physics , organic chemistry , astronomy , gene
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 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. © 1999 John Wiley & Sons, Inc.