A formula for the Theta invariant from Heegaard diagrams

TL;DR

This paper presents a combinatorial formula for the Theta invariant of rational homology spheres using explicit propagators derived from Heegaard diagrams, linking configuration space integrals with topological invariants.

Contribution

It introduces explicit Morse propagators associated with Heegaard diagrams to compute the Theta invariant combinatorially, connecting geometric and algebraic approaches.

Findings

Derived explicit propagators from Heegaard diagrams.

Proved a combinatorial formula for the Theta invariant.

Connected the Theta invariant to classical invariants like Casson-Walker and Pontrjagin classes.

Abstract

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.