Link invariants from racks

Friedrich Wagemann (University of Nantes)

Monday 3rd March, 2014 16:00-17:00 Maths 204


This is a report on joint work with Alissa Crans (LMU).

Given a codimension 2 embedding, i.e. a link  L:M\subset Q, in a
connected manifold Q (which we will suppose framed and transversally
oriented), J.H.C. Whitehead associates in 1949 to it the invariant

pi_2(Q,Q_0)  -> pi_1(Q_0),

where Q_0 = \overline{ Q\setminus N(M) } is the link complement.
The Whitehead invariant turns out to be a crossed module of groups. Later
Conway-Wraith '59, Joyce '82, Mateev '84 and Fenn-Rourke '92 refine this
invariant to a rack, called the fundamental rack. A rack is a generalization
of a group - one retains in its axiomatization only the
self-distributivity condition
which is fulfilled by the conjugation in a group.
Together with A. Crans, we investigate the notion of a crossed module of
We show as an application of our study that in the situation of a covering
pi: P -> Q
with compatible links L_Q : M\subset Q and L_P : pi^{-1}(M)\subset P, the
racks of L_Q and L_P fit together to give a crossed module of racks.

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