Inventiones mathematicae

, Volume 133, Issue 3, pp 613–681

A sum formula for the Casson-Walker invariant

  • Christine Lescop

DOI: 10.1007/s002220050256

Cite this article as:
Lescop, C. Invent math (1998) 133: 613. doi:10.1007/s002220050256


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 LA of the map from H1(∂A;Q) to H1(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 H1(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 H2(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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Christine Lescop
    • 1
  1. 1.Département de Mathématiques, Institut Fourier, UMR 5582, CNRS, BP 74, F-38402 Saint-Martin d'Hères Cedex, France (e-mail: