  For rational homology 3-spheres, there exist two universal finite-type
invariants: the Le-Murakami-Ohtsuki invariant and the
Kontsevich-Kuperberg-Thurston invariant. These invariants take values in the
same space of "Jacobi diagrams", but it is not known whether they are equal. In
2004, Lescop proved that the KKT invariant satisfies some "splitting formulas"
which relate the variations of KKT under replacement of embedded rational
homology handlebodies by others in a "Lagrangian-preserving" way. We show that
the LMO invariant satisfies exactly the same relations. The proof is based on
the LMO functor, which is a generalization of the LMO invariant to the category
of 3-dimensional cobordisms, and we generalize Lescop's splitting formulas to
this setting.
