Crossed modules of racks

Preprint, Other literature type English OPEN
Crans, Alissa S. ; Wagemann, Friedrich (2014)
  • Publisher: International Press of Boston
  • Journal: (issn: 1532-0073)
  • Subject: rack | link invariant | 2-group | Crossed module of groups | augmented rack | rack space | 20L05 | 57M25 | 18D05 | 57M10 | 18D35 | crossed module of racks | 2-rack | 57Q45 | 18B40, 18D05, 18D35, 20L05, 57M10, 57M25, 57Q45 | 18B40 | fundamental rack of a link | Mathematics - Quantum Algebra | covering of rack space

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 rack-space 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.
  • References (14)
    14 references, page 1 of 2

    [1] J. Baez, A. Lauda, Higher-Dimensional Algebra V: 2-Groups. Theory Theory and Applications of Categories 12 (2004), 423 - 491.

    [2] J. S. Carter, A. Crans, M. Elhamdadi, M. Saito, Unpublished correspondence

    [3] M. Forrester-Barker, Group objects and internal categories, available as math.CT/0212065.

    [4] R. Fenn, C. Rourke, Racks and links in codimension two. J. Knot Theory Ramifications 1 (1992), no. 4, 343 -406

    [5] R. Fenn, C. Rourke, B. Sanderson, Trunks and Classifying Spaces. Appl. Cat. Str. 3 (1995) 321-356

    [6] C. Kassel, Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995

    [7] L. H. Kauffman, J.F. Martins. Invariants of welded virtual knots via crossed module invariants of knotted surfaces. Compos. Math. 144 (4) (2008) 1046 -1080.

    [8] J.-L. Loday, Spaces with finitely many nontrivial homotopy groups. J. Pure Appl. Algebra 24 (1982), no. 2, 179 - 202.

    [9] J.F. Martins, Categorical Groups, Knots and Knotted Surfaces. J. Knot Theory Ramifications 16 No. 9, (2007) 1181 - 1217.

    [10] J.F. Martins, The Fundamental Crossed Module of the Complement of a Knotted Surface, Trans. Amer. Math. Soc. 361 (2009), 4593 - 4630.

  • Metrics
    No metrics available
Share - Bookmark