Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

TL;DR

This paper establishes a connection between reflexive and involutive Hochschild homologies and equivariant Loday constructions, providing a unified homotopical framework under certain algebraic conditions.

Contribution

It demonstrates that these homology theories can be realized as homotopy groups of an equivariant Loday construction, extending the understanding of their algebraic and topological structures.

Findings

Homology groups identified with equivariant Loday construction homotopy groups

Results hold when 2 is invertible and the abelian group is flat

Includes a relative version for associative algebras over arbitrary rings

Abstract

For associative rings with anti-involution several homology theories exists, for instance reflexive homology as studied by Graves and involutive Hochschild homology defined by Fern\`andez-Val\`encia and Giansiracusa. We prove that the corresponding homology groups can be identified with the homotopy groups of an equivariant Loday construction of the one-point compactification of the sign-representation evaluated at the trivial orbit, if we assume that $2$ is invertible and if the underlying abelian group of the ring is flat. We also show a relative version where we consider an associative $k$-algebra with an anti-involution where $k$ is an arbitrary ground ring.