Finite groups, spherical 2-categories, and 4-manifold invariants

Details Finite groups, spherical 2-categories, and 4-manifold invariants Finite groups, spherical 2-categories, and 4-manifold invariants

1999-02-28

arxiv.org/abs/math/9903003v3

Adv. Math. 153(2) (2000), 353-390

Mathematics

Quantum Algebra

LaTeX 44 pages, 10 eps.files, accepted for publication in Adv Math. Much better introduction explaining the fundamental ideas,

Scientific paper

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.

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