Monopoles and Landau-Ginzburg Models

Abstract

In this thesis, we define the monopole Floer homology for any pair (𝑌, 𝜔), where 𝑌 is any oriented compact 3-manifold with toroidal boundary and 𝜔 is a suitable closed 2-form on 𝑌 , generalizing the construction of Kronheimer-Mrowka for closed 3-manifolds. The basic setup is borrowed from the seminal paper of Meng-Taubes. This thesis will be divided into three parts:

∙ Part I is concerned with the geometry of planar ends. We exploit the framework of the gauged Landau-Ginzburg models to address two model problems for the (perturbed) Seiberg-Witten moduli spaces on either C x Σ or H²₊ x Σ, where Σ is any compact Riemann surface of genus ≥ 1. These results will lead eventually to the compactness theorem in the second part;

∙ In Part II, we supply the analytic foundation for this Floer theory based on the results from Part I. The Euler characteristic of this Floer homology recovers the Milnor-Turaev torsion invariant of 𝑌 by a classical theorem of Meng-Taubes and Turaev.

∙ In Part III, more topological properties of this Floer theory are explored in the special case that the boundary ∂𝑌 is disconnected and the 2-form 𝜔 is nonvanishing on ∂𝑌 . Using Floer’s excision theorem, we establish a gluing result for this Floer homology when two such 3-manifolds are glued suitably along their common boundary. As applications, we construct the monopole Floer 2-functor and the generalized cobordism maps. Using results of Kronheimer-Mrowka and Ni, we prove that for any such irreducible 𝑌 , this Floer homology detects the Thurston norm on 𝐻₂(𝑌, ∂𝑌; R) and the fiberness of 𝑌 . Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3-manifold.

This thesis is the compilation of the three arxiv preprints [Wan20a][Wan20b][Wan20c].