Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Mati\'c gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.Comment: 86 pages, 54 figures, to appear in Geometry and Topolog

The blob complex

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.Comment: 106 pages. Version 3 contains many improvements following suggestions from the referee and others, and some additional materia

Finite type invariants and fatgraphs

We define an invariant $\nabla_G(M)$ of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder $S\times I$, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, $\nabla_G$ is the composition with the $\iota_n$ maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., $\nabla_G$ establishes an isomorphism from an appropriate vector space $\bar{H}$ of homology cylinders to a certain algebra of Jacobi diagrams. Via composition $\nabla_{G'}\circ\nabla_G^{-1}$ for any pair of fatgraph spines G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmuller space, as a group of automorphisms of this algebra. The space $\bar{H}$ comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how $\nabla_G$ interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of $\nabla_G$.Comment: 39 page

A diagrammatic approach to categorification of quantum groups III

We categorify the idempotented form of quantum sl(n).Comment: 88 pages, LaTeX2e with xypic and pstricks macros, 3 eps file