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Invariants de type fini des variétés de dimension trois et structures spinorielles

Abstract : M. Goussarov and K. Habiro have introduced in the middle Nineties a finite type invariants theory for compact oriented 3-manifolds. In this thesis, the Goussarov-Habiro theory is refined to the cases when those manifolds are equipped with spin structures or complex spin structures. For closed 3-manifolds with spin structures, we geometrically characterize degree 0 invariants by revealing the role played by some quadratic forms. We also prove that Rochlin invariant of spin 3-manifolds is a finite type invariant of degree 1. We are then interested in homology cylinders over a compact oriented surface with 0 or 1 boundary component. By using the spin refinement of Goussarov-Habiro theory, we characterize degree 1 invariants of homology cylinders. For closed 3-manifolds with complex spin structures, we give a geometric characterization of degree 0 invariants. The latter makes use of a quadratic function canonically associated to any closed oriented 3-manifold endowed with a complex spin structure. Finally, we calculate how the Abelian Reidemeister-Turaev torsion of a closed 3-manifold with complex spin structure changes, when the manifold is twisted along a closed connected splitting surface by a diffeomorphism which acts trivially in homology. In particular, we deduce that, in a restricted sense, the Abelian Reidemeister-Turaev torsion is multiplicatively a finite type invariant of degree 1
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Contributor : Gwénaël Massuyeau <>
Submitted on : Tuesday, November 5, 2002 - 2:02:06 PM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 6:15:30 PM


  • HAL Id : tel-00001919, version 1



Gwénaël Massuyeau. Invariants de type fini des variétés de dimension trois et structures spinorielles. Mathématiques [math]. Université de Nantes, 2002. Français. ⟨tel-00001919⟩



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