A formula for the Theta invariant from Heegaard diagrams

Christine Lescop (1)

1 IF - Institut Fourier

The Theta invariant is the simplest 3-manifold invariant defined with configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing X in the 2-point configuration space of a Q-sphere M. These propagators represent the linking form of M so that $\Theta(M,X)$ can be thought of as the cube of the linking form of M with respect to the combing X. The Theta invariant is the sum of $6 \lambda(M)$ and $p_1(X)/4$, where $\lambda$ denotes the Casson-Walker invariant, and $p_1$ is an invariant of combings that is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these ''Morse propagators'', constructed with Greg Kuperberg, to prove a combinatorial formula for the Theta invariant in terms of Heegaard diagrams.

Mots clés: Theta invariant; Heegaard diagrams; Heegaard splittings; Casson-Walker invariant; configuration space integrals; finite type invariants of 3-manifolds; homology spheres; combings; perturbative expansion of Chern-Simons theory; invariant Theta; diagrammes de Heegaard; scindements de Heegaard; invariant de Casson-Walker; intégrales d'espaces de configurations; invariants de type fini des variétés de dimension 3; sphères d'homologie; développement perturbatif de la théorie de Chern-Simons

Domaines: Topologie géométrique [math.GT]

Christine Lescop. A formula for the Theta invariant from Heegaard diagrams. 2014. ⟨hal-00732323v2⟩