Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

E-infinity structure in hyperoctahedral homology

Pages: 1 – 19

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a1


Daniel Graves (School of Mathematics, University of Leeds, United Kingdom)


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.


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

2010 Mathematics Subject Classification

13D03, 55N35, 55N45, 55S12

Copyright © 2023, Daniel Graves. Permission to copy for private use granted.

Received 23 August 2021

Received revised 3 January 2022

Accepted 13 January 2022

Published 1 March 2023