Homology, Homotopy and Applications

Volume 16 (2014)

Number 2

Crossed modules of racks

Pages: 85 – 106

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a5

Authors

Alissa S. Crans (Mathematics Department, Loyola Marymount University, Los Angeles, California, U.S.A.)

Friedrich Wagemann (Laboratoire de Mathématiques Jean Leray, Université de Nantes, France)

Abstract

We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the relation between crossed modules of groups and strict 2-groups. Then we explore topological applications. We show that by applying the rackspace functor, a crossed module of racks gives rise to a covering. Our main result shows how the fundamental racks associated to links upstairs and downstairs in a covering fit together to form a crossed module of racks.

Keywords

crossed module of groups, rack, augmented rack, crossed module of racks, rack space, covering of rack space, link invariant, fundamental rack of a link, 2-group, 2-rack

2010 Mathematics Subject Classification

18B40, 18D05, 18D35, 20L05, 57M10, 57M25, 57Q45

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