The Århus integral of rational homology 3-spheres III: Relation with the Le–Murakami–Ohtsuki invariant

Abstract.

Continuing the work started in [Å-I] and [Å-II], we prove the relationship between the Århus integral and the invariant Ω (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in [LMO]. The basic reason for the relationship is that both constructions afford an interpretation as “integrated holonomies”. In the case of the Århus integral, this interpretation was the basis for everything we did in [Å-I] and [Å-II]. The main tool we used was “formal Gaussian integration”. For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map j _m , as “formal negative dimensional integration”. The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.