The Tambara-Yamagami categories and 3-manifold invariants

Abstract

We prove that if two Tambara-Yamagami categories TY(A,\chi,u) and TY(A',\chi',u') give rise to the same state sum invariants of 3-manifolds and the order of one of the groups A, A' is odd, then u=u' and there is a group isomorphism A\approx A' carrying \chi to \chi'. The proof is based on an explicit computation of the state sum invariants for the lens spaces of type (k,1).