2004/09/30 by Jérôme Dubois, Jerome Dubois · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M05 #msc:57M25 #msc:57M27
paper · pdf · doi:10.2140/agt.2006.6.373
published as Algebr. Geom. Topol. 6 (2006) 373-404 · This is the version published by Algebraic & Geometric Topology on 12 March 2006
openalex publication_date 2006/03/12 · arxiv created 2009/03/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Abstract. For a knot K in S 3, 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 the group of K (see Section 3). We next prove that this volume form is a topological knot invariant (see Section 4) and explore some of its properties (see Section 5). Motivation and Main ideas In 1985, A. Casson constructed an integer valued invariant of integral homology 3-spheres. The original definition of the Casson invariant is based on SU(2)representation spaces. Informally speaking, the Casson invariant of an homology 3-sphere M counts algebraically the number of conjugacy classes of irreducible SU(2)-representations of π1(M) in the same sense that the Lefschetz number of a map counts the number of fixed points, see [AM90] or [GM92]. In 1992, X.-S. Lin used an analogue Casson’s original construction to define an integer knot invariant. He indirectly proved that this invariant is equal to half the signature of knots, see [Lin92]. At first sight, this equality between to apparently different algebricogeometric