E-infinity structure in hyperoctahedral homology

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.

Keywords

hyperoctahedral homology, crossed simplicial group, $E_\infty$-algebra, Dyer–Lashof operation, Pontryagin product

2010 Mathematics Subject Classification

13D03, 55N35, 55N45, 55S12

Full Text (PDF format)

Received 23 August 2021

Received revised 3 January 2022

Accepted 13 January 2022

Published 1 March 2023