Topological symmetric and braid homologies

Abstract: 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.