2024/09/26 by Gabriel Angelini‐Knoll, Angelini-Knoll, Gabriel, Mona Merling +3
Computer Science · Mathematics · Medicine · #16E40 #18N60 #55P42 #55P43 #55P91 #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #K-Theory and Homology (math.KT) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2409.18187
openalex publication_date 2024/09/26 · openalex created_date 2024/10/27 · openalex updated_date 2026/07/28
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 S1-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 C4-action, which carries a Pin(2)-action. We prove that THQ of a loop space with twisted C4-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.