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- The universal sl(3)-link homologyPublication . Mackaay, Marco; Vaz, PedroWe define the universal sl(3)-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl(3)-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new. Following an approach similar to Gornik's we show that this new link homology can be described in terms of Khovanov's original sl(2)-link homology.
- The foam and the matrix factorization sl(3) link homologies are equivalentPublication . Mackaay, Marco; Vaz, PedroWe prove that the universal rational sl(3) link homologies which were constructed by Khovanov in [3] and the authors in [7], using foams, and by Khovanov and Rozansky in [4], using matrix factorizations, are naturally isomorphic as projective functors from the category of links and link cobordisms to the category of bigraded vector spaces.
- 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).
- Bar-Natan's Khovanov homology for coloured linksPublication . Mackaay, Marco; Turner, PaulUsing Bar-Natan's link homology we define a homology theory for framed links whose components are labelled by irreducible representations of the group U-q(sl(2)). We then compute this explicitly.
- The foam and the matrix factorization sl3 link homologies are equivalentPublication . Mackaay, Marco; Vaz, PedroWe prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.
- 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.
- sl(N)-link homology (N >= 4) using foams and the Kapustin-Li formulaPublication . Mackaay, Marco; Stosic, Marko; Vaz, PedroWe use foams to give a topological construction of a rational link homology categorifying the sl(N) link invariant, for N >= 4. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky [9]. We show that for any link our homology is isomorphic to the Khovanov-Rozansky [11] homology.