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We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in CY, and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants Br1,Br2 using techniques developed in the case of the semi-simple sub-quotient of Rep(U q (sl 2 )) (q a principal 4r th root of unity) by Roberts Ro1. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter Y6 and Roberts Ro2.
Crane et al. (Tue,) studied this question.