Invariants of Walker and Lescop

Abstract

The Casson invariant, originally defined for integral homology 3-spheres, was extended by Walker [302] to rational homology 3-spheres. Partial extensions were also proposed by Boyer and Lines [35] and Boyer and Nicas [36]. Walker defined his invariant by extending Casson’s SU(2) intersection theory to include reducible representations, which arise as long as the first integral homology of the rational homology sphere does not vanish. Most remarkably, Walker’s invariant also admits a purely combinatorial definition in terms of surgery presentations. This definition was later generalized by Lescop [182], who defined an invariant for all oriented closed 3-manifolds thus extending both Casson’s and Walker’s invariants.