For a knot in the 3-sphere and a regular representation of its group 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 . In another way, we construct using Casson’s original construction a natural volume form on the SU(2)-representation space of . Next, we compare these two apparently different points of view on the representation variety and finally prove that they produce the same topological knot.
Etant donné un noeud dans la sphère tridimensionnelle et une représentation régulière de son groupe dans SU(2), on construit une forme torsion de Reidemeister non abélienne sur le premier groupe de cohomologie tordue de l’extérieur de . Cette forme torsion de Reidemeister permet de définir une forme volume sur l’espace de représentations de 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 dans SU(2). On établit enfin que ces deux points de vue en apparence distincts produisent en fait le même invariant topologique de noeuds.
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)
@article{AIF_2005__55_5_1685_0, author = {Dubois, J\'er\^ome}, title = {Non abelian {Reidemeister} torsion and volume form on the {SU(2)-representation} space of knot groups}, journal = {Annales de l'Institut Fourier}, pages = {1685--1734}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {5}, year = {2005}, doi = {10.5802/aif.2136}, mrnumber = {2172277}, zbl = {1077.57009}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2136/} }
TY - JOUR AU - Dubois, Jérôme TI - Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups JO - Annales de l'Institut Fourier PY - 2005 SP - 1685 EP - 1734 VL - 55 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2136/ DO - 10.5802/aif.2136 LA - en ID - AIF_2005__55_5_1685_0 ER -
%0 Journal Article %A Dubois, Jérôme %T Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups %J Annales de l'Institut Fourier %D 2005 %P 1685-1734 %V 55 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2136/ %R 10.5802/aif.2136 %G en %F AIF_2005__55_5_1685_0
Dubois, Jérôme. Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups. Annales de l'Institut Fourier, Volume 55 (2005) no. 5, pp. 1685-1734. doi : 10.5802/aif.2136. http://www.numdam.org/articles/10.5802/aif.2136/
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