On Quinn's Invariants of 2-dimensional CW-complexes

Abstract

Given a semisimple stable autonomous tensor category over a field $K$, to any group presentation with finite number of generators we associate an element $Q(P)\in K$ invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm of Frank Quinn. The new definition allows us to present a relatively simple proof of the invariance and to evaluate $Q(P)$ for some presentations. On the basis of some numerical calculations over different Gelfand-Kazhdan categories, we make a conjecture which relates the value of $Q(P)$ for two different classes of presentations.