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Torsion de Reidemeister non abélienne et forme volume sur l'espace des représentations du groupe d'un noeud

Abstract : For a knot $K$ in $S^3$, we construct according to Casson -- or more precisely taking into account Lin's (cf. J. Differential Geom. 35 (1992) 337-357) and Heusener's (cf. Topology Appl. 127 (2003) 175-197) further works -- a volume form on the $SU(2)$-representation space of the group $G_K$ of the knot $K$. More precisely, let us denote by $\mathrm(Reg)(K)$ the set of $SO(3)$-conjugacy classes of \emph(regular) representations of $G_K$ into $SU(2)$, then $\mathrm(Reg)(K)$ is a one-dimensional manifold and we prove that $\mathrm(Reg)(K)$ also carries a natural $1$-volume form. Next, we show how to interpret this volume form as a non abelian Reidemeister torsion. Finally, we give an explicit computation of this volume form for torus knots and fibered knots and we also compute the Reidemeister torsion of the Brieskorn homology spheres with coefficients in the adjoint representation. Furthermore, we study the behavior of this volume form under mutation.
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https://tel.archives-ouvertes.fr/tel-00003782
Contributor : Jerome Dubois <>
Submitted on : Tuesday, November 18, 2003 - 10:30:34 PM
Last modification on : Tuesday, May 7, 2019 - 6:30:09 PM
Long-term archiving on: : Friday, April 2, 2010 - 7:04:34 PM

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  • HAL Id : tel-00003782, version 1

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Jérôme Dubois. Torsion de Reidemeister non abélienne et forme volume sur l'espace des représentations du groupe d'un noeud. Mathématiques [math]. Université Blaise Pascal - Clermont-Ferrand II, 2003. Français. ⟨tel-00003782⟩

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