Tangles, relative character varieties, and holonomy perturbed traceless flat moduli spaces

TL;DR

This paper demonstrates that the restriction map from a perturbed SU(2) traceless flat moduli space of a tangle to its boundary is a Lagrangian immersion, using Weinstein category composition and Hamiltonian isotopies.

Contribution

It establishes the Lagrangian property of the restriction map in the context of holonomy perturbations and boundary moduli spaces, introducing new techniques involving Weinstein categories.

Findings

Restriction map is a Lagrangian immersion.

Holonomy perturbations induce Hamiltonian isotopies.

The moduli space of (S^2,4) is self-identical.

Abstract

We prove that the restriction map from the subspace of regular points of the holonomy perturbed SU(2) traceless flat moduli space of a tangle in a 3-manifold to the traceless flat moduli space of its boundary marked surface is a Lagrangian immersion. A key ingredient in our proof is the use of composition in the Weinstein category, combined with the fact that SU(2) holonomy perturbations in a cylinder induce Hamiltonian isotopies. In addition, we show that $(S^2,4)$, the 2-sphere with four marked points, is its own traceless flat SU(2) moduli space.