Une nouvelle définition de l'invariant de Casson

In [4], following [2], we have defined an invariant $\text{Δ(f)∈}\text{Q}$, for any $\text{f∈}\text{M}{}_{\mathrm{g,1}}$, the mapping class group of a compact, connected, oriented surface with connected boundary, genus g. For $\text{f∈}\text{T}{}_{\mathrm{g,1}}$ (a certain subgroup of the Torelli group), we have shown in [4], using Casson surgery formula, that Δ(f) co??ncides with the Casson invariant [1] of the homology sphere M_f, obtained by gluing two handlebodies along f. The purpose of this Note is to prove directly (i.e., without reference to Casson) that Δ(f), for $\text{f∈}\text{T}{}_{\mathrm{g,1}}$, depends only on M_f. The surgery formula, which is a difficult point in Casson version, follows almost immediatly from the definition of Δ(f). To cite this article: B. Perron, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 199–204.