- Authors:
- DOI:
- 10.1016/j.aim.2010.04.008
- 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 .
- Type:
- Journal article
- Language:
- English
- Published in:
- Advances in Mathematics, 2010, Vol 225, Issue 4, p. 2117-2161
- Main Research Area:
- Science/technology
- Publication Status:
- Published
- Review type:
- Peer Review
- Submission year:
- 2010
- Scientific Level:
- Scientific
- ID:
- 122491971
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