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<p>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.</p>

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The homology of permutation racks

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月26日

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  • Victoria Lebed,
  • Markus Szymik

摘要

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.