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.





Original language | English |
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Journal | Advances in Mathematics |
Volume | 225 |
Issue number | 4 |
Pages (from-to) | 2117-2161 |
Number of pages | 45 |
ISSN | 0001-8708 |
DOIs | |
Publication status | Published - 2010 |
Externally published | Yes |