Reflexive homology

TL;DR

Reflexive homology extends Hochschild homology to include involutions, relating to $C_2$-equivariant homology of free loop spaces, with computational tools and invariance properties explored.

Contribution

This paper introduces reflexive homology, linking it to equivariant homology, provides computational methods, and establishes its properties and relations to existing theories.

Findings

Reflexive homology of a group algebra is isomorphic to $C_2$-equivariant homology of free loop space.

A bicomplex for computing reflexive homology is developed.

Reflexive homology satisfies Morita invariance.

Abstract

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fern\`andez-Val\`encia and Giansiracusa coincides with reflexive homology.