Splitting formulas for the LMO invariant of rational homology three–spheres

Abstract

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.