The invariant Θ 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 λ by the formula
Θ(M,X)=6λ(M)+p1​(X)/4, where p1​ is an invariant of combings
that is simply related to a Gompf invariant. In [arXiv:1209.3219], we proved a
combinatorial formula for the Θ-invariant in terms of Heegaard diagrams,
equipped with decorations that define combings, from the definition of Θ
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 6λ(M) and p1​(X)/4 for integral homology spheres, by proving surgery
formulae both for the combinatorial invariant and for p1​.Comment: 63 page