On the traceless SU(2) character variety of the 6-punctured 2-sphere

Abstract

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of CP3 , branched over the singular Kummer surface, with the branch locus in R(S2,6) corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover Fn−1→S2 branched over 2n points, combined with the theorem of Narasimhan–Ramanan which identifies R(F2) with CP3 . The singular points of R(S2,6) correspond to abelian representations, and we prove that each has a neighborhood in R(S2,6) homeomorphic to a cone on S2×S3 .