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Fixed point sets and the fundamental group I: semi-free actions on G-CW-complexes

Published online by Cambridge University Press:  03 August 2023

Sylvain Cappell Open the ORCID record for Sylvain Cappell [Opens in a new window] ,
Shmuel Weinberger  and
Min Yan Open the ORCID record for Min Yan [Opens in a new window]
Sylvain Cappell
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA (cappell@cims.nyu.edu)
Shmuel Weinberger
Affiliation:
University of Chicago, Chicago, IL, USA (shmuel@math.uchicago.edu)
Min Yan
Affiliation:
Hong Kong University of Science and Technology, Hong Kong, People's Republic of China (mamyan@ust.hk)
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Abstract

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

Keywords

group actionsalgebraic K-theorySmith theory

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 6 , December 2024 , pp. 1639 - 1660
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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