2026-09-21T13:11:19Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1392572022-01-15T03:17:01Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Mrowka, Tomasz
Wang, Donghao
Massachusetts Institute of Technology. Department of Mathematics
2022-01-14T14:59:52Z
2022-01-14T14:59:52Z
2021-06
2021-05-25T12:47:46.498Z
https://hdl.handle.net/1721.1/139257
0000-0001-9554-9511
In this thesis, we define the monopole Floer homology for any pair (π, π), where π is any oriented compact 3-manifold with toroidal boundary and π is a suitable closed 2-form on π , 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:
β 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 Ξ£ or HΒ²β x Ξ£, where Ξ£ is any compact Riemann surface of genus β₯ 1. These results will lead eventually
to the compactness theorem in the second part;
β 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 π by a classical theorem of Meng-Taubes and Turaev.
β In Part III, more topological properties of this Floer theory are explored in the special case that the boundary βπ is disconnected and the 2-form π is nonvanishing on βπ . Using Floerβs 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 π , this Floer homology detects the Thurston norm on π»β(π, βπ; R) and the fiberness of π . 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.
Massachusetts Institute of Technology
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Monopoles and Landau-Ginzburg Models
Thesis
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Doctoral
Doctor of Philosophy
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<Abstract>In this thesis, we define the monopole Floer homology for any pair (π, π), where π is any oriented compact 3-manifold with toroidal boundary and π is a suitable closed 2-form on π , 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:
β 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 Ξ£ or HΒ²β x Ξ£, where Ξ£ is any compact Riemann surface of genus β₯ 1. These results will lead eventually
to the compactness theorem in the second part;
β 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 π by a classical theorem of Meng-Taubes and Turaev.
β In Part III, more topological properties of this Floer theory are explored in the special case that the boundary βπ is disconnected and the 2-form π is nonvanishing on βπ . Using Floerβs 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 π , this Floer homology detects the Thurston norm on π»β(π, βπ; R) and the fiberness of π . 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>
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