Topological $ΔG$ homology of rings with twisted $G$-action
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arXiv
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| Natura: | Preprint |
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2024
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| author | Angelini-Knoll, Gabriel Merling, Mona Péroux, Maximilien |
| author_facet | Angelini-Knoll, Gabriel Merling, Mona Péroux, Maximilien |
| contents | We construct topological $ΔG$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18187 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological $ΔG$ homology of rings with twisted $G$-action Angelini-Knoll, Gabriel Merling, Mona Péroux, Maximilien Algebraic Topology K-Theory and Homology 16E40, 55P43, 55P91, 18N60, 55P42 We construct topological $ΔG$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action. |
| title | Topological $ΔG$ homology of rings with twisted $G$-action |
| topic | Algebraic Topology K-Theory and Homology 16E40, 55P43, 55P91, 18N60, 55P42 |
| url | https://arxiv.org/abs/2409.18187 |