A sum formula for the Casson-Walker invariant

Abstract.

We study the following question: How does the Casson-Walker invariant λ of a rational homology 3-sphere obtained by gluing two pieces along a surface depend on the two pieces? Our partial answer may be stated as follows. For a compact oriented 3-manifold A with boundary ∂A, the kernel L _A of the map from H ₁(∂A;Q) to H ₁(A;Q) induced by the inclusion is called the Lagrangian of A. Let Σ be a closed oriented surface, and let A, A′, B and B′ be four rational homology handlebodies such that ∂A, ∂A′, −∂B and −∂B′ are identified via orientation-preserving homeomorphisms with Σ. Assume that L _A = L _{A ′} and L _B = L _{B ′} inside H ₁(Σ;Q) and also assume that L _A and L _B are transverse. Then we express

in terms of the form induced on ∧³ L _A by the algebraic intersection on H ₂(A∪_Σ−A′) paired to the analogous form on ∧³ L _B via the intersection form of Σ. The simple formula that we obtain naturally extends to the extension of the Casson-Walker invariant of the author. It also extends to gluings along non-connected surfaces.