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# Surfaces et invariants de type fini en dimension 3

Abstract : The main topic of this work is the theory of finite type invariants of homology 3-spheres, developed by Ohtsuki, Le, Garoufalidis, Goussarov, Habiro and others. In a first part, the variation of a degree 2n invariant after surgery along a surface with respect to an element of the 2n-th term of the lower central series of the Torelli group is studied. For a commutator of 2n elements of the Torelli group, this variation is expressed in terms of the Johnson homomorphism of the 2n elements and of the weight system of the invariant.

The clasper calculus of Goussarov and Habiro gives topological equivalences between surgeries on embedded handlebodies in 3-manifolds. This calculus has already been crucial in order to describe the behaviour of the finite type invariants under many topological modifications. In the second part, the clasper calculus is refined. This
refinement allows us to give a geometric surgery formula for the degree 4 invariants. The variation of a degree $4$ invariant after surgery on a knot is expressed in terms of invariants of embedded curves in a regular neighbourhood of a Seifert surface of the knot.
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00113863
Contributor : Emmanuel Auclair Connect in order to contact the contributor
Submitted on : Tuesday, November 14, 2006 - 6:03:20 PM
Last modification on : Friday, March 25, 2022 - 11:09:51 AM
Long-term archiving on: : Friday, November 25, 2016 - 1:13:15 PM

### Identifiers

• HAL Id : tel-00113863, version 1

### Citation

Emmanuel Auclair. Surfaces et invariants de type fini en dimension 3. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2006. Français. ⟨NNT : ⟩. ⟨tel-00113863⟩

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