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For a finite group$$G$$of not prime power order, Oliver showed that the obstruction for a finite CW-complex$$F$$to be the fixed point set of a contractible finite$$G$$-CW-complex is determined by the Euler characteristic$$\chi (F)$$. (He also has similar results for compact Lie group actions.) We show that the analogous problem for$$F$$to be the fixed point set of a finite$$G$$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on$$K_0$$[2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.
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Fixed point sets and the fundamental group I: semi-free actions on G -CW-complexes
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.
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- Award ID(s):
- 2105451
- PAR ID:
- 10505193
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- ISSN:
- 0308-2105
- Page Range / eLocation ID:
- 1 to 22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/prm.2023.63
