{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139257"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139257","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"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].","abstract_html":"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].","abstract_has_math":false,"creators":["Wang, Donghao"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Mrowka, Tomasz"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:22:03Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/139257","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Mrowka, Tomasz"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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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]."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Monopoles and Landau-Ginzburg Models"]}]}],"canonical_facts":{"dc:contributor.advisor":["Mrowka, Tomasz"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Wang, Donghao"],"dc:date.accessioned":["2022-01-14T14:59:52Z"],"dc:date.available":["2022-01-14T14:59:52Z"],"dc:date.issued":["2021-06"],"dc:description.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]."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139257"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Monopoles and Landau-Ginzburg Models"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:22:03Z"}