Real Hochschild homology as an equivariant Loday construction

TL;DR

This paper explores how equivariant Loday constructions can interpret Real Hochschild homology of discrete rings, using geometric actions of dihedral and symmetric groups to analyze isotropy subgroups.

Contribution

It introduces a novel interpretation of Real Hochschild homology as an equivariant Loday construction involving dihedral and symmetric group actions.

Findings

Real Hochschild homology can be modeled as an equivariant Loday construction.

Geometric actions of dihedral groups on polygons are used to analyze isotropy.

Examples include actions of symmetric groups on permutohedra's 1-skeleta.

Abstract

Equivariant Loday constructions are a means for providing geometric interpretations of equivariant homology theories. They are usually constructed for a simplicial $G$-set and a $G$-Tambara functor. We study situations where -- depending on the isotropy subgroups occurring in the simplicial $G$-set -- one can work with $H$-Tambara functors for a suitable subgroup $H$ of $G$. We apply this to give an interpretation of Real Hochschild homology of discrete $E_\sigma$-rings as equivariant Loday constructions where we consider $2m$-gons with a geometrically defined action of the dihedral groups $D_{2m}$ for all $m \geq 1$. The action of symmetric groups on $1$-skeleta of permutohedra also gives examples with isotropy groups $C_2$.