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

### Guillem Cazassus, Chris Herald and Paul Kirk

Vol. 5 (2022), No. 1, 95–121
##### 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 $\left({S}^{2},4\right)$, the 2-sphere with four marked points, is its own traceless flat SU(2) moduli space.

##### Keywords
holonomy perturbation, Lagrangian immersion, Floer homology, flat moduli space, traceless character variety, quilted Floer homology
##### Mathematical Subject Classification
Primary: 57K18, 57K31, 57R58
Secondary: 81T13