Isovariant homotopy theoy and fixed point invariants

Inbar Klang; Sarah Yeakel

doi:10.1016/j.aim.2023.109298

Isovariant homotopy theoy and fixed point invariants

Inbar Klang, Sarah Yeakel

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein–Williams equivariant intersection theory for a finite group G. We prove that under certain reasonable dimension and codimension conditions on H-fixed subspaces (for ), the fixed points of a self-map of a compact smooth G-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.

Original language	English
Article number	109298
Pages (from-to)	1-34
Number of pages	34
Journal	Advances in Mathematics
Volume	433
Early online date	20 Sept 2023
DOIs	https://doi.org/10.1016/j.aim.2023.109298
Publication status	Published - 15 Nov 2023

Funding

This project was inspired by conversations at a Women in Topology workshop at the Mathematical Sciences Research Institute. We would like to thank Kate Ponto and Cary Malkiewich for suggesting the project and providing support throughout. We would also like to thank Michael Mandell for an illuminating discussion on Theorem 3.10 , and the anonymous referee whose comments greatly improved the quality of this paper. We would like to thank Peter Haine for introducing us to the notion of weak model categories, and Maru Sarazola for an enlightening talk and useful conversations about fibrantly induced model structures. Some of this paper is based upon work supported by the National Science Foundation under Grant No. 1440140 , while the first-named author and the second-named author were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the semester of Fall 2022 and the Spring 2020 semester, respectively. This project was inspired by conversations at a Women in Topology workshop at the Mathematical Sciences Research Institute. We would like to thank Kate Ponto and Cary Malkiewich for suggesting the project and providing support throughout. We would also like to thank Michael Mandell for an illuminating discussion on Theorem 3.10, and the anonymous referee whose comments greatly improved the quality of this paper. We would like to thank Peter Haine for introducing us to the notion of weak model categories, and Maru Sarazola for an enlightening talk and useful conversations about fibrantly induced model structures. Some of this paper is based upon work supported by the National Science Foundation under Grant No. 1440140, while the first-named author and the second-named author were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the semester of Fall 2022 and the Spring 2020 semester, respectively.

Funders	Funder number
Michael Mandell
Peter Haine
National Science Foundation	1440140

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10.1016/j.aim.2023.109298

https://www.sciencedirect.com/science/article/pii/S0001870823004413

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title = "Isovariant homotopy theoy and fixed point invariants",

abstract = "An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein–Williams equivariant intersection theory for a finite group G. We prove that under certain reasonable dimension and codimension conditions on H-fixed subspaces (for ), the fixed points of a self-map of a compact smooth G-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.",

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AB - An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein–Williams equivariant intersection theory for a finite group G. We prove that under certain reasonable dimension and codimension conditions on H-fixed subspaces (for ), the fixed points of a self-map of a compact smooth G-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we study isovariant maps between manifolds up to isovariant homotopy, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.

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DO - 10.1016/j.aim.2023.109298

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JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109298

ER -

Isovariant homotopy theoy and fixed point invariants

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