Finite type invariants and fatgraphs |
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Authors: | Jø rgen Ellegaard Andersen,Jean-Baptiste Meilhan |
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Affiliation: | a Department of Mathematics, Center for the Topology and Quantization of Moduli Spaces, Aarhus University, DK-8000 Aarhus C, Denmark b Departments of Mathematics, University of Southern California, Los Angeles, CA 90089, USA c Departments of Mathematics and Physics/Astronomy, University of Southern California, Los Angeles, CA 90089, USA d Institut Fourier, Université Grenoble 1, 38402 St Martin d'Hères, France |
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Abstract: | We define an invariant ∇G(M) of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder Σ×I, Σ is a connected surface with at least one boundary component, and G is a fatgraph spine of Σ. In effect, ∇G is the composition with the ιn 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., ∇G establishes an isomorphism from an appropriate vector space of homology cylinders to a certain algebra of Jacobi diagrams. Via composition for any pair of fatgraph spines G,G′ of Σ, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmüller space, as a group of automorphisms of this algebra. The space 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 ∇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 ∇G. |
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Keywords: | Finite type invariants Fatgraphs Mapping class groups Teichmü ller spaces Homology cylinders |
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