A multiplicative version of the tom Dieck splitting
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916720102342656 |
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| author | Blumberg, Andrew J. Mandell, Michael A. |
| author_facet | Blumberg, Andrew J. Mandell, Michael A. |
| contents | While the classical tom Dieck splitting in equivariant stable homotopy theory is typically regarded as a formula for suspension spectra in the genuine equivariant stable category, it can be interpreted as a calculation of the fixed points of $G$-spectra that are derived pushforwards from the naive equivariant stable category. We then establish a corresponding multiplicative splitting formula for derived pushforwards of $N_{\infty}$ ring spectra. Just as the usual tom Dieck splitting characterizes the equivariant stable category associated to an $N_{\infty}$ operad $\mathcal{N}$, the multiplicative tom Dieck splitting characterizes the $G$-symmetric monoidal structure on the genuine equivariant stable category associated to $\mathcal{N}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_02721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A multiplicative version of the tom Dieck splitting Blumberg, Andrew J. Mandell, Michael A. Algebraic Topology Primary 55P43, 55P91 While the classical tom Dieck splitting in equivariant stable homotopy theory is typically regarded as a formula for suspension spectra in the genuine equivariant stable category, it can be interpreted as a calculation of the fixed points of $G$-spectra that are derived pushforwards from the naive equivariant stable category. We then establish a corresponding multiplicative splitting formula for derived pushforwards of $N_{\infty}$ ring spectra. Just as the usual tom Dieck splitting characterizes the equivariant stable category associated to an $N_{\infty}$ operad $\mathcal{N}$, the multiplicative tom Dieck splitting characterizes the $G$-symmetric monoidal structure on the genuine equivariant stable category associated to $\mathcal{N}$. |
| title | A multiplicative version of the tom Dieck splitting |
| topic | Algebraic Topology Primary 55P43, 55P91 |
| url | https://arxiv.org/abs/2505.02721 |