A topological quantum field theory for the Le-Murakami-Ohtsuki invariant of three-dimensional manifolds
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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) representations) of a subgroup [Special characters omitted.] <math> <f> <sc>L<inf><mit>g</mit></inf></sc></f> </math> of the Mapping Class Group that contains the Torelli group. The N = 1 truncation produces a TQFT for the Casson-Walker-Lescop invariant. This construction aims to shed some light to the questions of topological interpretation of quantum invariants of manifolds and of determining the structure of the Mapping Class Group using quantum invariants.