Non Abelian Reidemeister Torsion and Volume Form on the Su(2)-representation Space of Knot Groups

For a knot K in S and a regular representation ρ of its group GK 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 GK (see Section 5). In another way, we construct according to Casson— or more precisely taking into account Lin’s [Lin92] and Heusener’s [Heu03] further works—a volume form on the SU(2)-representation space of GK (see Section 6). Next, we compare these two apparently different points of view— the first by means of the Reidemeister torsion and the second defined “à la Casson”—and finally prove that they produce the same topological knot invariant (see Section 7). Introduction: Motivation and Main ideas The space of conjugacy classes of irreducible SU(2)-representations of a knot group is a real semi-algebraic set and its regular part is a one-dimensional manifold (for more details see infra). The aim of this paper is to compare two a priori different constructions on this one-dimensional manifold: one by mean of Reidemeister torsion and another using Casson’s original construction. Firstly, we associate to any regular SU(2)-representation ρ of the knot group a non abelian Reidemeister torsion form on the first cohomology group of the knot exterior with coefficients in the adjoint representation associated to ρ. Secondly, we construct “à la Casson” a volume form on the one-dimensional manifold consisting of conjugacy classes of regular representations, which appears to be a knot invariant. Finally, we prove that these a priori two different points of view are indeed equivalent (see Theorem 7.1). The Reidemeister torsion was introduced for the first time in 1935 by K. Reidemeister in his work [Rei35] on the combinatorial classification of 3-dimensional lens spaces. The torsion is a combinatorial invariant which is not a homotopy invariant. It is became a fundamental tool in low-dimensional topology, see for example the Turaev’s monograph [Tur02]. Informally speaking, the Reidemeister torsion is a graded version of the determinant in the same way that the Euler characteristic is a graded version of the dimension. In 1985, A. Casson constructed an integer valued invariant of integral homology 3-spheres which appeared extremely useful. The original definition of the Casson Date: 9th June 2008. 1991 Mathematics Subject Classification. 57M25; 57Q10; 57M27.