Torsion in one-term distributive homology

Volume 225 / 2014

Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra Fundamenta Mathematicae 225 (2014), 75-94 MSC: 55N35, 18G60. DOI: 10.4064/fm225-1-5


The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn–Rourke–Sanderson [FRS] and Carter–Kamada–Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type $(n,1)$ and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, we show that if a shelf satisfies a certain, rather general, condition then the one-term homology is trivial—this answers a conjecture from [Prz] affirmatively.


  • Alissa S. CransDepartment of Mathematics
    Loyola Marymount University
    Los Angeles, CA 90045, U.S.A.
  • Józef H. PrzytyckiDepartment of Mathematics
    George Washington University
    Washington, DC 20052, U.S.A.
    Institute of Mathematics
    University of Gdańsk
    80-952 Gdańsk, Poland
  • Krzysztof K. PutyraDepartment of Mathematics
    Columbia University
    New York, NY 10027, U.S.A.

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