Additive power operations in equivariant cohomology
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912846537818112 |
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| author | Bonventre, Peter J. Guillou, Bertrand J. Stapleton, Nathaniel J. |
| author_facet | Bonventre, Peter J. Guillou, Bertrand J. Stapleton, Nathaniel J. |
| contents | Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any $G$-space $X$ and positive integer $m$, we give an explicit description of the smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times BΣ_m)$ for which the reduced $m$th power operation $\underline{E}^0(X) \to \underline{E}^0(X \times BΣ_m )/\underline{J}$ is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of $G\timesΣ_m$-Green functors. This theorem also specializes to characterize the appropriate ideal $\underline{J}$ when $E$ is a $G_\infty$-ring in global spectra. We give example computations for the sphere spectrum, complex $K$-theory, and Morava $E$-theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_11078 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Additive power operations in equivariant cohomology Bonventre, Peter J. Guillou, Bertrand J. Stapleton, Nathaniel J. Algebraic Topology 55N91, 55S25 Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any $G$-space $X$ and positive integer $m$, we give an explicit description of the smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times BΣ_m)$ for which the reduced $m$th power operation $\underline{E}^0(X) \to \underline{E}^0(X \times BΣ_m )/\underline{J}$ is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of $G\timesΣ_m$-Green functors. This theorem also specializes to characterize the appropriate ideal $\underline{J}$ when $E$ is a $G_\infty$-ring in global spectra. We give example computations for the sphere spectrum, complex $K$-theory, and Morava $E$-theory. |
| title | Additive power operations in equivariant cohomology |
| topic | Algebraic Topology 55N91, 55S25 |
| url | https://arxiv.org/abs/2001.11078 |