A formula for the $\Theta$–invariant from Heegaard diagrams

Abstract

The $\Theta$–invariant is the simplest $3$–manifold invariant defined by 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 $\mathbb{Q}$–sphere $M$. These propagators represent the linking form of $M$ so that $\Theta \left(M,X\right)$ can be thought of as the cube of the linking form of $M$ with respect to the combing $X$. The invariant $\Theta$ is the sum of $6\lambda \left(M\right)$ and $p_1\left(X\right)∕4$, where $\lambda$ denotes the Casson–Walker invariant, and $p_1$ is an invariant of combings, which 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.