A formula for the Theta invariant from Heegaard diagrams

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 $\mathrm{\Theta }(M,X)$ 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 $p\mathrm{\_}1(X)\mathrm{/}4$, where $\lambda$ denotes the Casson-Walker invariant, and $p\mathrm{\_}1$ 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.