Topological symmetric and braid homologies
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917522519883776 |
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| author | Angelini-Knoll, Gabriel Chan, David Gerhardt, Teena Merling, Mona Péroux, Maximilien |
| author_facet | Angelini-Knoll, Gabriel Chan, David Gerhardt, Teena Merling, Mona Péroux, Maximilien |
| contents | We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $Δ\mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22946 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological symmetric and braid homologies Angelini-Knoll, Gabriel Chan, David Gerhardt, Teena Merling, Mona Péroux, Maximilien Algebraic Topology Category Theory K-Theory and Homology 16E40, 55P43, 57T30, 18M75, 18N60, 55P42 We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $Δ\mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies. |
| title | Topological symmetric and braid homologies |
| topic | Algebraic Topology Category Theory K-Theory and Homology 16E40, 55P43, 57T30, 18M75, 18N60, 55P42 |
| url | https://arxiv.org/abs/2605.22946 |