Event № 902

In the nineties, M. Kontsevich defined an invariant Z of integral homology spheres equipped with a parallelization. This invariant is a graded object so that it makes sense to ask for a description of its lowest degree part, called the Theta-invariant. In 2000, G. Kuperberg and D. Thurston revisited the construction of Z and proved many of its properties. In particular they proved that Theta(M, tau) splits as the sum of 6 times the Casson invariant of M and 1/4 of the first relative Pontrjagin class. Later on, C. Lescop gave another construction of Z and proved that if M is a rational homology sphere and X a homotopy class of nowhere vanishing vector fields on M, then Theta(M, X) is defined and satisfies a relation as above and provides an invariant of nowhere vanishing vector fields. 

In this talk, I will present a purely combinatorial proof of the latter relation, using finite type invariant theory and C. Lescop's recent combinatorial formula for the Theta-invariant from Heegaard diagrams. All prerequisites will be reviewed.