The homology of permutation racks

TL;DR

This paper computes the complete integral homology of permutation racks using homotopical algebra and spectral sequences, providing a new comprehensive understanding of their algebraic topology.

Contribution

It introduces a method employing homotopical algebra and spectral sequences to fully calculate the homology of permutation racks, a previously unresolved problem.

Findings

Spectral sequence degenerates for all permutation racks

Complete integral homology computed for all permutation racks

Method demonstrates potential for broader applications in rack homology

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 equivariant homology, and for which we show that it always degenerates. The blueprint given in this paper demonstrates the high potential for further exploitation of these techniques.