Equivariant cohomology for cyclic groups of square-free order
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916218845265920 |
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| author | Basu, Samik Ghosh, Surojit |
| author_facet | Basu, Samik Ghosh, Surojit |
| contents | The main objective of this paper is to compute $RO(G)$-graded cohomology of $G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We compute these groups for the constant Mackey functor $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, we show that the groups $\underline{H}^α_G(S^0)$ are mostly determined by the fixed point dimensions of the virtual representations $α$, except in the case of $\underline{A}$ coefficients when the fixed point dimensions of $α$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_09669 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Equivariant cohomology for cyclic groups of square-free order Basu, Samik Ghosh, Surojit Algebraic Topology The main objective of this paper is to compute $RO(G)$-graded cohomology of $G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We compute these groups for the constant Mackey functor $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, we show that the groups $\underline{H}^α_G(S^0)$ are mostly determined by the fixed point dimensions of the virtual representations $α$, except in the case of $\underline{A}$ coefficients when the fixed point dimensions of $α$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes. |
| title | Equivariant cohomology for cyclic groups of square-free order |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2006.09669 |