Homology, Homotopy and Applications

Volume 7 (2005)

Number 1

Extensions of racks and quandles

Pages: 151 – 167

DOI: https://dx.doi.org/10.4310/HHA.2005.v7.n1.a8


Nicholas Jackson (Mathematics Institute, University of Warwick, Coventry, United Kingdom)


A rack is a set equipped with a bijective, self-right-distributive binary operation, and a quandle is a rack which satisfies an idempotency condition.

In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [9] and the more general one given by Andruskiewitsch and Graña [1]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [3], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles.

We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [6, 7] as special cases.


racks, quandles, extensions, modules, homology, cohomology

2010 Mathematics Subject Classification

18E10, 18G15, 18G60

Published 1 January 2005