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Finite type invariants and fatgraphs

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  • Jørgen Ellegaard Andersen, Denmark
  • Alex Bene, Denmark
  • Jean-Baptiste Odet Thierry Meilhan, Denmark
  • Robert Penner, Denmark
We define an invariant backward differenceG(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, backward differenceG 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., backward differenceG establishes an isomorphism from an appropriate vector space View the MathML source of homology cylinders to a certain algebra of Jacobi diagrams. Via composition View the MathML source 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 View the MathML source 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 backward differenceG 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 backward differenceG.
Original languageEnglish
JournalAdvances in Mathematics
Pages (from-to)2117-2161
Number of pages45
Publication statusPublished - 2010

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