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

Gwénaël Massuyeau

Algebraic & Geometric Topology 14 (2014) 3553–3588

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.

$3$–manifold, finite-type invariant, LMO invariant, Lagrangian-preserving surgery
Mathematical Subject Classification 2010
Primary: 57M27
Received: 11 October 2013
Accepted: 15 April 2014
Published: 15 January 2015
Gwénaël Massuyeau
Institut de Recherche Mathématique Avancée
Université de Strasbourg and CNRS
7 rue René Descartes
67084 Strasbourg