Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Dubois, Jérôme

doi:10.5802/aif.2136

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups
[Torsion de Reidemeister non abélienne et forme volume sur l'espace des représentations du groupe d'un noeud dans SU(2)]

Dubois, Jérôme ¹

¹ Université de Genève, section de mathématiques, CP64, 2-4 rue du lièvre, 1211 Genève 4 (Suisse), Université Blaise Pascal, laboratoire de mathématiques, avenue des Landais, 63177 Aubière cedex (France)

Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1685-1734.

Résumé
Abstract

Etant donné un noeud $K$ dans la sphère tridimensionnelle et une représentation régulière de son groupe $G_{K}$ dans SU(2), on construit une forme torsion de Reidemeister non abélienne sur le premier groupe de cohomologie tordue de l’extérieur de $K$ . Cette forme torsion de Reidemeister permet de définir une forme volume sur l’espace de représentations de $G_{k}$ dans SU(2). D’un autre point de vue, en s’inspirant de la construction originale de l’invariant de Casson, on construit une forme volume naturelle sur l’espace de représentations de $G_{K}$ dans SU(2). On établit enfin que ces deux points de vue en apparence distincts produisent en fait le même invariant topologique de noeuds.

For a knot $K$ in the 3-sphere and a regular representation of its group $G_{K}$ into SU(2) we construct a non abelian Reidemeister torsion form on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion form provides a volume form on the SU(2)-representation space of $G_{K}$ . In another way, we construct using Casson’s original construction a natural volume form on the SU(2)-representation space of $G_{K}$ . Next, we compare these two apparently different points of view on the representation variety and finally prove that they produce the same topological knot.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2136

Classification : 57M25, 57Q10, 57M27
Keywords: Knot groups, representation space, volume form, Reidemeister torsion, Casson invariant, adjoint representation, SU(2)
Mot clés : groupe de noeuds, espace de représentations, forme volume, torsion de Reidemeister, invariant de Casson, représentation adjointe, SU(2)

Affiliations des auteurs :

Dubois, Jérôme ¹

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Dubois, Jérôme. Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1685-1734. doi : 10.5802/aif.2136. https://www.numdam.org/articles/10.5802/aif.2136/

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