  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 $\nabla_G$.
