Fixed Point Sets and the Fundamental Group I: Semi-free Actions on G-CW-Complexes

2020/10/28 by Sylvain Cappell, Shmuel Weinberger, Cappell, Sylvain +3
Mathematics · #57Q91 #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:57Q91

paper · pdf · doi:10.48550/arxiv.2010.14987

26 pages

arxiv created 2022/02/18 · arxiv updated 2022/02/21

Abstract

Smith theory says that the fixed point of a semi-free action of a group G on a contractible space is \bb Z_p-acyclic for any prime factor p of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain K-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the K-theoretical obstruction.

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