2024/09/08 by Siqi He, He, Siqi, Richard Wentworth +3 · 3 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2409.04956
This paper studies the relationship between an analytic compactification of the moduli space of flat SL2(ℂ) connections on a closed, oriented 3-manifold M defined by Taubes, and the Morgan-Shalen compactification of the SL2(ℂ) character variety of the fundamental group of M. We exhibit an explicit correspondence between ℤ/2 harmonic 1-forms, measured foliations, and equivariant harmonic maps to ℝ-trees, as initially proposed by Taubes. As an application, we prove that ℤ/2 harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular ℤ/2 harmonic 1-forms but have compact SL2(ℂ) character varieties, which resolves a folklore conjecture.