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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.13971 |
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| _version_ | 1866912076048367616 |
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| author | Beaudry, Agnès Lewis, Chloe May, Clover Pauli, Sabrina Tatum, Elizabeth |
| author_facet | Beaudry, Agnès Lewis, Chloe May, Clover Pauli, Sabrina Tatum, Elizabeth |
| contents | This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(ΠB)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations.
We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(ΠB)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Guide to Equivariant Parametrized Cohomology Beaudry, Agnès Lewis, Chloe May, Clover Pauli, Sabrina Tatum, Elizabeth Algebraic Topology 55N25 This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(ΠB)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(ΠB)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$. |
| title | A Guide to Equivariant Parametrized Cohomology |
| topic | Algebraic Topology 55N25 |
| url | https://arxiv.org/abs/2410.13971 |