Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles

Abstract

The traceless SU(2) character variety $R(S^2,\{a_i,b_i{\}}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the SU(2) character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension $4n-6$.

For generic holonomy perturbations $\pi$, the traceless SU(2) character variety $R_{\pi }(Y,L)$ of an $n$-stranded tangle $L$ in a homology 3-ball $Y$ is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to $R(S^2,\{a_i,b_i{\}}_{i=1}^n)$ is a Lagrangian immersion which preserves the cone neighborhood structure near the singular stratum.

For generic holonomy perturbations $\pi$, the variant $R_{\pi }^{♮}(Y,L)$, obtained by taking the connected sum of $L$ with a Hopf link and considering SO(3) representations with $w_2$ supported near the extra component, is a smooth compact manifold without boundary of dimension $2n-3$, which Lagrangian immerses into the smooth stratum of $R(S^2,\{a_i,b_i{\}}_{i=1}^n)$.

The proofs of these assertions consist of stratified transversality arguments to eliminate non-generic strata in the character variety and to insure that the restriction map to the boundary character variety is also generic.

The main tool introduced to establish abundance of holonomy perturbations is the use of holonomy perturbations along curves $C$ in a cylinder $F\times I$, where $F$ is a closed surface. When $C$ is obtained by pushing an embedded curve on $F$ into the cylinder, we prove that the corresponding holonomy perturbation induces one of Goldman's generalized Hamiltonian twist flows on the SU(2) character variety $\mathcal{M}(F)$ associated to the curve $C$.