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- Holonomy and parallel transport for Abelian gerbesPublication . Mackaay, Marco; Picken, R.In this paper, we establish a one-to-one correspondence between U(1)-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group U(1) on a simply connected manifold M is a group morphism from the thin second homotopy group to U(1), satisfying a smoothness condition, where a homotopy between maps from [0,1](2) to M is thin when its derivative is of rank less than or equal to2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. (C) 2002 Elsevier Science (USA).
- Finite groups, spherical 2-categories, and 4-manifold invariantsPublication . Mackaay, MarcoIn this paper we dc line a class of state-sum invariants of closed oriented piece wise lineal 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 an earlier paper. We show that the state survival invariants of Birmingham and Ratowski, who Studied Dijkgraaf Witten type invariants in dimension 3, 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, wt indicate how our construction is related to homotopy 3-types. This connection suggests that there ale many more interesting examples of our construction to be found in the work on homotopy 3-types, by Brown, For example. (C) 2000 Academic press.