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Un nouvel invariant pour les sphères d’homologie de dimension trois [d’après Casson]. (A new invariant of homology 3-spheres (after Casson)). (French) Zbl 0674.57013

Sémin. Bourbaki, 40ème Année, Vol. 1987/88, Exp. No. 693, Astérisque 161-162, 151-164 (1988).
[For the entire collection see Zbl 0659.00006.]
The author reports on certain (currently unpublished) work of Casson. Casson defined an integer invariant for 3-dimensional closed manifolds H, which are homology spheres. This invariant, \(\lambda\) (H), was reinterpreted by C. H. Taubes [Casson’s invariant for homology 3- spheres and Fredholm-Euler class, Abstr. Am. Math. Soc. 7, 188 (1986); Gauge theory on asymptotically periodic 4-manifolds, J. Differ. Geom. 25, 363-430 (1987; Zbl 0615.57009)] and appears in the “Floer homology”, [P. J. Braam, Floer homology for homology 3-spheres (Preprint, Oxford, 1988)]. Let S denote the set of diffeomorphism classes of oriented 3-dimensional manifolds with homology of the 3-sphere. Let \(\Delta_ K(t)\) denote the Alexander polynomial of the knot \(K\subset H\), \(H\in S\), normalized so that \(\Delta_ K(t)=\Delta_ K(t^{-1})\). Let \(K_ n\) denote a homology sphere obtained from the knot K by Dehn surgery of type 1/n. The principal result proved is the following: There exists a map \(\lambda\) from S to Z with the following properties: (1) \(\lambda (H)=0\) if every map from \(\pi_ 1(H)\) to SU(2) is trivial. (2) The Rohlin invariant of \(H=\lambda (H)\) mod 2. (3) \(\lambda (-H)=-\lambda (H).\) (4) \(\lambda (H_ 1\#H_ 2)=\lambda (H_ 1)+\lambda (H_ 2)\). (5) \(\lambda (K_{n+1})-\lambda (K_ n)=\Delta_ K''(1)\).
Reviewer: L.Neuwirth

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
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