E-infinity structure in hyperoctahedral homology

TL;DR

This paper establishes an E-infinity algebra structure on hyperoctahedral homology, enabling the definition of Dyer-Lashof operations and a Pontryagin product, revealing new algebraic properties and limitations of the homology theory.

Contribution

It introduces an E-infinity algebra structure on hyperoctahedral homology, allowing for new operations and insights into its algebraic nature and invariance properties.

Findings

Hyperoctahedral homology admits Dyer-Lashof operations.

A Pontryagin product makes it an associative, graded-commutative algebra.

Hyperoctahedral homology does not preserve Morita equivalence.

Abstract

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.