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**Notes on the Casson invariant of three dimensional homology spheres.
(Notes sur l’invariant de Casson des sphères d’homologie de dimension trois.)**
*(French)*
Zbl 0776.57008

In 1985 Casson introduced a new \(Z\)-valued invariant of \(Z\)-homology 3- spheres, which was defined in terms of the representations of the fundamental group in SU(2) and which reduced modulo (2) to the Rokhlin invariant. Analytic interpretations of this invariant lead to the development of Floer homology and the invariant has since been extended to a family of invariants for all 3-manifolds, related to representations of \(\pi_ 1\) in other compact semisimple Lie groups.

This paper presents a detailed exposition of Casson’s original approach. The development is in part axiomatized: standard techniques are used to show that (up to rescaling) there is at most one function from homology 3-spheres to the integers whose values change in a prescribed fashion when the argument is modified by Dehn surgery on a knot or a 2-component boundary link. The algebraic intersection of the varieties of representations of the free groups associated to a Heegaard splitting of the 3-manifold gives rise to such a function. However, the proof that it is well behaved with respect to Dehn surgery on 2-component boundary links requires a hard theorem of P. E. Newstead [Topology 6, 241- 262 (1967; Zbl 0201.234)]. The authors are careful to point out where this result is needed and to what extent it can be avoided.

There are three appendices. The first reviews connections between the Alexander polynomial and the Robertello invariant. The other two (by Marin and Lescop, respectively) give elementary calculations of the Casson invariant for special classes of homology spheres, those obtained by Dehn surgeries on a knot with an unknotted Seifert surface of genus 1, and those which are Seifert fibred with 3 exceptional fibres.

This paper presents a detailed exposition of Casson’s original approach. The development is in part axiomatized: standard techniques are used to show that (up to rescaling) there is at most one function from homology 3-spheres to the integers whose values change in a prescribed fashion when the argument is modified by Dehn surgery on a knot or a 2-component boundary link. The algebraic intersection of the varieties of representations of the free groups associated to a Heegaard splitting of the 3-manifold gives rise to such a function. However, the proof that it is well behaved with respect to Dehn surgery on 2-component boundary links requires a hard theorem of P. E. Newstead [Topology 6, 241- 262 (1967; Zbl 0201.234)]. The authors are careful to point out where this result is needed and to what extent it can be avoided.

There are three appendices. The first reviews connections between the Alexander polynomial and the Robertello invariant. The other two (by Marin and Lescop, respectively) give elementary calculations of the Casson invariant for special classes of homology spheres, those obtained by Dehn surgeries on a knot with an unknotted Seifert surface of genus 1, and those which are Seifert fibred with 3 exceptional fibres.

Reviewer: J.A.Hillman (Sydney)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |