, Volume 133, Issue 3, pp 613-681

A sum formula for the Casson-Walker invariant

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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(AΣA′) 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.

Oblatum 6-III-1995 & 31-X-1997