Spherical 2-categories and 4-manifold invariants

Abstract

In this paper I define the notion of a non-degenerate finitely semi-simple semi-strict spherical 2-category of non-zero dimension. Given such a 2-category I define a state-sum for any triangulated compact closed oriented 4-manifold and show that this state-sum does not depend on the chosen triangulation by proving invariance under the 4D Pachner moves. This invariant of piece-wise linear closed compact oriented 4-manifolds generalizes the Crane-Yetter invariant and probably it is also a generalization of the Crane-Frenkel invariant. As an example we show how to obtain a 2-category of the right kind from a finite group and a 4-cocycle on this group. The invariant we obtain from this example looks like a four dimensional version of the Dijkgraaf-Witten invariant.