1. Topological $\Delta G$ homology of rings with twisted $G$-action
- Author
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Angelini-Knoll, Gabriel, Merling, Mona, and Péroux, Maximilien
- Subjects
Mathematics - Algebraic Topology ,Mathematics - K-Theory and Homology ,16E40, 55P43, 55P91, 18N60, 55P42 - Abstract
We provide a unifying framework for studying variants of topological Hochschild homology such as Real topological Hochschild homology. Associated to a crossed simplicial group $\Delta G$, a category that generalizes Connes' cyclic category, we introduce an invariant of rings with twisted $G$-action, which we call topological $\Delta G$-homology. Here a ring with twisted $G$-action is a common generalization of a ring with involution and a ring with $G$-action. Our construction provides homotopical analogues of quaternionic homology, symmetric homology, braid homology, and hyperoctahedral homology defined by Fiedorowicz--Loday. In the quaternionic case our construction, called quaternionic topological Hochschild homology, is equipped with a left Pin(2)-action and we compute it for loop spaces with a twisted $C_4$-action. As an important step along the way, we introduce a new family of crossed simplicial groups called the twisted symmetric crossed simplicial groups and we prove that they corresponds to operads for algebras with twisted $G$-action., Comment: 61 pages. Comments welcome!
- Published
- 2024