The homology of permutation racks

Abstract

Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still very few examples of racks whose homology has been fully calculated. In this paper, we compute the entire integral homology of all permutation racks. Our method of choice involves homotopical algebra, which was brought to bear on the homology of racks only recently. For our main result, we establish a spectral sequence, which reduces the problem to one in Borel’s equivariant homology, and for which we show that it always degenerates. The blueprint given in this paper demonstrates a high potential for further exploitation of these techniques.

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Item Type:	Article
Authors/Creators:	Lebed, V. Szymik, M. https://orcid.org/0000-0002-7078-2699
Copyright, Publisher and Additional Information:	© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Permutations; Racks; Rack homology; Equivariant Borel homology; Spectral sequences
Dates:	Accepted: 23 May 2025 Published (online): 26 June 2025 Published: 26 June 2025
Institution:	The University of Sheffield
Academic Units:	The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematical and Physical Sciences
Depositing User:	Symplectic Sheffield
Date Deposited:	27 Jun 2025 10:06
Last Modified:	27 Jun 2025 10:06
Status:	Published
Publisher:	Springer Science and Business Media LLC
Refereed:	Yes
Identification Number:	10.1007/s00209-025-03797-5
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:228449

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The homology of permutation racks

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