Hyperoctahedral Homology for Involutive Algebras

TL;DR

This paper introduces hyperoctahedral homology for involutive algebras, linking it to equivariant stable homotopy theory and providing new insights into the homology of group algebras with involution.

Contribution

It establishes a connection between hyperoctahedral homology and equivariant stable homotopy theory, expanding the understanding of involutive algebra homology.

Findings

Hyperoctahedral homology relates to equivariant infinite loop space homology.

For groups of odd order, the homology of the group algebra matches fixed point homology.

The paper provides a new homological perspective on involutive algebra structures.

Abstract

Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group.