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 CP 3 , branched over the singular Kummer surface, with the branch locus in R(S 2 , 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 → S 2 branched over 2n points, combined with the theorem of Narasimhan-Ramanan which identifies R(F2) with CP 3 . The singular points of R(S 2 , 6) correspond to abelian representations, and we prove that each has a neighborhood in R(S 2 , 6) homeomorphic to a cone on S 2 × S 3 .2010 Mathematics Subject Classification. 57M05, 53D30.

Bordered theory for pillowcase homology

“…After the pillowcase R(S 2 , 4), the next space of interest is R(S 2 , 6). It is already a complicated singular 6-dimensional manifold, see [17]. See also [13] for the study of R(S 2 , 2n).…”

The moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points

“…The structure of M s ( È 1 , 6) for µ = 1/4, corresponding to the traceless character variety, was recently described by Kirk [9]. For an elliptic curve X, it is straightforward to show that for small weight we have…”

The correspondence induced on the pillowcase by the earring tangle

“…However, in our setting, neither the $$R(F, 2k)$$'s nor the $$R(Y_i,T_i)$$'s are smooth manifolds, due to the even number of punctures. Nevertheless, it may be possible to develop Floer theory in $$R(S^2, 2k)$$'s, via equivariant techniques, or by only considering the smooth stratum (see [45] for the description of these spaces in general, and [53] for a detailed study of $$R(S^2,6)$$). The Floer field theory framework would result in a well‐defined generalized Lagrangian invariant inside the pillowcase $$\begin{equation*} \text{1pt}\xrightarrow {\underline{L}(D^3,T)} R(S^2,4)=P.$$…”