Fixed point sets and the fundamental group II: Euler characteristics

Sylvain Cappell; Shmuel Weinberger; Min Yan

doi:10.1017/prm.2023.75

Fixed point sets and the fundamental group II: Euler characteristics

Sylvain Cappell, Shmuel Weinberger, Min Yan

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For a finite group of not prime power order, Oliver showed that the obstruction for a finite CW-complex to be the fixed point set of a contractible finite -CW-complex is determined by the Euler characteristic. (He also has similar results for compact Lie group actions.) We show that the analogous problem for to be the fixed point set of a finite -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

Original language	English (US)
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs	https://doi.org/10.1017/prm.2023.75
State	Accepted/In press - 2023

Keywords

Euler characteristic
Smith theory
algebraic K-theory
fixed points
group actions
transformation groups

ASJC Scopus subject areas

General Mathematics

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10.1017/prm.2023.75

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title = "Fixed point sets and the fundamental group II: Euler characteristics",

abstract = "For a finite group of not prime power order, Oliver showed that the obstruction for a finite CW-complex to be the fixed point set of a contractible finite -CW-complex is determined by the Euler characteristic. (He also has similar results for compact Lie group actions.) We show that the analogous problem for to be the fixed point set of a finite -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.",

keywords = "Euler characteristic, Smith theory, algebraic K-theory, fixed points, group actions, transformation groups",

author = "Sylvain Cappell and Shmuel Weinberger and Min Yan",

note = "Publisher Copyright: Copyright {\textcopyright} The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.",

year = "2023",

doi = "10.1017/prm.2023.75",

language = "English (US)",

journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",

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T1 - Fixed point sets and the fundamental group II

T2 - Euler characteristics

AU - Cappell, Sylvain

AU - Weinberger, Shmuel

AU - Yan, Min

PY - 2023

Y1 - 2023

N2 - For a finite group of not prime power order, Oliver showed that the obstruction for a finite CW-complex to be the fixed point set of a contractible finite -CW-complex is determined by the Euler characteristic. (He also has similar results for compact Lie group actions.) We show that the analogous problem for to be the fixed point set of a finite -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

AB - For a finite group of not prime power order, Oliver showed that the obstruction for a finite CW-complex to be the fixed point set of a contractible finite -CW-complex is determined by the Euler characteristic. (He also has similar results for compact Lie group actions.) We show that the analogous problem for to be the fixed point set of a finite -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

KW - Euler characteristic

KW - Smith theory

KW - algebraic K-theory

KW - fixed points

KW - group actions

KW - transformation groups

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JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ER -

Fixed point sets and the fundamental group II: Euler characteristics

Abstract

Keywords

ASJC Scopus subject areas

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