Monopoles and Landau-Ginzburg Models
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 semina...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2022
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| Online Access: | https://hdl.handle.net/1721.1/139257 |
| _version_ | 1826209775215443968 |
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| author | Wang, Donghao |
| author2 | Mrowka, Tomasz |
| author_facet | Mrowka, Tomasz Wang, Donghao |
| author_sort | Wang, Donghao |
| collection | MIT |
| description | 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]. |
| first_indexed | 2024-09-23T14:29:44Z |
| format | Thesis |
| id | mit-1721.1/139257 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T14:29:44Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1392572022-01-15T03:17:01Z Monopoles and Landau-Ginzburg Models Wang, Donghao Mrowka, Tomasz Massachusetts Institute of Technology. Department of Mathematics 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]. Ph.D. 2022-01-14T14:59:52Z 2022-01-14T14:59:52Z 2021-06 2021-05-25T12:47:46.498Z Thesis https://hdl.handle.net/1721.1/139257 0000-0001-9554-9511 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Wang, Donghao Monopoles and Landau-Ginzburg Models |
| title | Monopoles and Landau-Ginzburg Models |
| title_full | Monopoles and Landau-Ginzburg Models |
| title_fullStr | Monopoles and Landau-Ginzburg Models |
| title_full_unstemmed | Monopoles and Landau-Ginzburg Models |
| title_short | Monopoles and Landau-Ginzburg Models |
| title_sort | monopoles and landau ginzburg models |
| url | https://hdl.handle.net/1721.1/139257 |
| work_keys_str_mv | AT wangdonghao monopolesandlandauginzburgmodels |