Parametrised Poincaré duality and equivariant fixed points methods
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913369795067904 |
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| author | Hilman, Kaif Kirstein, Dominik Kremer, Christian |
| author_facet | Hilman, Kaif Kirstein, Dominik Kremer, Christian |
| contents | In this article, we introduce and develop the notion of parametrised Poincaré duality in the formalism of parametrised higher category theory by Martini-Wolf, in part generalising Cnossen's theory of twisted ambidexterity to the nonpresentable setting. We prove several basechange results, allowing us to move between different coefficient categories and ambient topoi. We then specialise the general framework to yield a good theory of equivariant Poincaré duality spaces for compact Lie groups and apply our basechange results to obtain a suite of isotropy separation methods. Finally, we employ this theory to perform various categorical Smith-theoretic manoeuvres to prove, among other things, a generalisation of a theorem of Atiyah-Bott and Conner-Floyd on group actions with single fixed points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parametrised Poincaré duality and equivariant fixed points methods Hilman, Kaif Kirstein, Dominik Kremer, Christian Algebraic Topology Geometric Topology In this article, we introduce and develop the notion of parametrised Poincaré duality in the formalism of parametrised higher category theory by Martini-Wolf, in part generalising Cnossen's theory of twisted ambidexterity to the nonpresentable setting. We prove several basechange results, allowing us to move between different coefficient categories and ambient topoi. We then specialise the general framework to yield a good theory of equivariant Poincaré duality spaces for compact Lie groups and apply our basechange results to obtain a suite of isotropy separation methods. Finally, we employ this theory to perform various categorical Smith-theoretic manoeuvres to prove, among other things, a generalisation of a theorem of Atiyah-Bott and Conner-Floyd on group actions with single fixed points. |
| title | Parametrised Poincaré duality and equivariant fixed points methods |
| topic | Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2405.17641 |