A combinatorial definition of the Theta-invariant from Heegaard diagrams

TL;DR

This paper provides a combinatorial formula for the Theta-invariant of rational homology 3-spheres with combings, proving it matches the sum of six times the Casson-Walker invariant and a combing invariant, without using configuration spaces.

Contribution

It introduces a purely combinatorial method to compute the Theta-invariant, establishing its equivalence to known invariants and extending previous results to pairs (M,X).

Findings

The combinatorial formula defines an invariant of pairs (M,X).

The invariant equals 6 times the Casson-Walker invariant plus p_1(X)/4.

Surgery formulas are proved for both the combinatorial invariant and p_1.

Abstract

The invariant $\Theta$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $\lambda$ by the formula $\Theta(M,X)=6\lambda(M)+p_1(X)/4$, where $p_1$ is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a combinatorial formula for the $\Theta$-invariant in terms of Heegaard diagrams, equipped with decorations that define combings, from the definition of $\Theta$ as an algebraic intersection in a configuration space. In this article, we prove that this formula defines an invariant of pairs $(M,X)$ without referring to configuration spaces, and we prove that this invariant is the sum of $6 \lambda(M)$ and $p_1(X)/4$ for integral homology spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$.