1999/03/31 by Marco Mackaay
Mathematics · #math.QA #math.CT
published as Adv. Math. 153(2) (2000), 353-390 · LaTeX 44 pages, 10 eps.files, accepted for publication in Adv Math. Much better introduction explaining the fundamental ideas, equivalence relation on weak mon. 2-structures defined and theorem added
arxiv created 1999/11/09 · arxiv updated 2009/11/30
In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in a previous paper, although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski, who studied Dijkgraaf-Witten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are non-trivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as by Brown, for example.