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

  • Christopher M. Herald

    University of Nevada, Reno, USA
  • Paul Kirk

    Indiana University, Bloomington, USA
Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles cover
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Abstract

The traceless SU(2) character variety R(S2,{ai,bi}i=1n)R(S^2,\{a_i,b_i\}_{i=1}^n) of a 2n2n-punctured 2-sphere is the symplectic reduction of a Hamiltonian nn-torus action on the SU(2) character variety of a closed surface of genus nn. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimension 4n64n-6.

For generic holonomy perturbations π\pi, the traceless SU(2) character variety Rπ(Y,L)R_\pi(Y,L) of an nn-stranded tangle LL in a homology 3-ball YY is stratified with a finite singular stratum and top stratum a smooth manifold. The restriction to R(S2,{ai,bi}i=1n)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 π\pi, the variant Rπ(Y,L)R_\pi^\natural(Y,L), obtained by taking the connected sum of LL with a Hopf link and considering SO(3) representations with w2w_2 supported near the extra component, is a smooth compact manifold without boundary of dimension 2n32n-3, which Lagrangian immerses into the smooth stratum of R(S2,{ai,bi}i=1n)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 CC in a cylinder F×IF\times I, where FF is a closed surface. When CC is obtained by pushing an embedded curve on FF 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 M(F)\mathcal{M}(F) associated to the curve CC.

Cite this article

Christopher M. Herald, Paul Kirk, Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles. Quantum Topol. 9 (2018), no. 2, pp. 349–418

DOI 10.4171/QT/110