Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Hyperoctahedral homology for involutive algebras

Pages: 1 – 26

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a1


Daniel Graves (School of Mathematics and Statistics, University of Sheffield, United Kingdom)


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.


hyperoctahedral homology, crossed simplicial group, functor homology, bar construction, equivariant infinite loop spaces

2010 Mathematics Subject Classification

13D03, 55N35, 55P47, 55U15

Copyright © 2021, Daniel Graves. Permission to copy for private use granted.

Received 16 November 2020

Accepted 25 January 2021

Published 3 November 2021