7 research outputs found
A functorial LMO invariant for Lagrangian cobordisms
Lagrangian cobordisms are three-dimensional compact oriented cobordisms
between once-punctured surfaces, subject to some homological conditions. We
extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor
from the category of Lagrangian cobordisms to a certain category of Jacobi
diagrams. We prove some properties of this functorial LMO invariant, including
its universality among rational finite-type invariants of Lagrangian
cobordisms. Finally, we apply the LMO functor to the study of homology
cylinders from the point of view of their finite-type invariants.Comment: 59 pages with many figures. Some minor changes in the writing, and a
few precisions adde
A TQFT associated to the LMO invariant of three-dimensional manifolds
We construct a Topological Quantum Field Theory (in the sense of Atiyah)
associated to the universal finite-type invariant of 3-dimensional manifolds,
as a functor from the category of 3-dimensional manifolds with parametrized
boundary, satisfying some additional conditions, to an algebraic-combinatorial
category. It is built together with its truncations with respect to a natural
grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The
TQFT(s) induce(s) a (series of) representation(s) of a subgroup of
the Mapping Class Group that contains the Torelli group. The N=1 truncation
produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been
considerably shorten, and section 3 has been slightly shorten, since they
will constitute a separate paper. Section 4, which contained only announce of
results, has been suprimated; it will appear in detail elsewhere.
Consequently some statements have been re-numbered. No mathematical changes
have been made.