The Tambara-Yamagami categories and 3-manifold invariants

Abstract

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

10 pages. In the second version the introduction has been considerably extended and the references updated