Results on spectral sequences for monopole and singular instanton Floer homologies
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2018
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| Online Access: | http://hdl.handle.net/1721.1/117864 |
| _version_ | 1826190357675638784 |
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| author | Gong, Sherry, Ph. D. Massachusetts Institute of Technology |
| author2 | Tomasz S. Mrowka. |
| author_facet | Tomasz S. Mrowka. Gong, Sherry, Ph. D. Massachusetts Institute of Technology |
| author_sort | Gong, Sherry, Ph. D. Massachusetts Institute of Technology |
| collection | MIT |
| description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. |
| first_indexed | 2024-09-23T08:39:03Z |
| format | Thesis |
| id | mit-1721.1/117864 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T08:39:03Z |
| publishDate | 2018 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1178642019-04-09T18:59:17Z Results on spectral sequences for monopole and singular instanton Floer homologies Gong, Sherry, Ph. D. Massachusetts Institute of Technology Tomasz S. Mrowka. Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics. Mathematics. Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. Cataloged from PDF version of thesis. Includes bibliographical references (pages 107-108). We study two gauge-theoretic Floer homologies associated to links, the singular instanton Floer homology introduced in [15] and the monopole Floer homology, which is explained in the book [16]. For both cases, we study in particular the spectral sequence that relates the Floer homologies to the Khovanov homologies of links. In our study of singular instanton Floer homology, we introduce a version of Khovanov homology for alternating links with marking data, W, inspired by singular instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in [15] for this marked Khovanov homology collapses on the E2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on w; thus, the instanton homology also does not depend on W for non-split alternating links. We study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on w. In our study of monopole Floer homology, we construct families of metrics on the cobordisms that are used to construct differentials in the spectral sequence relating the Khovanov homology of a link to the monopole Floer homology of its double branched cover, such that each metric has positive scalar curvature. This allows us to conclude that the Seiberg-Witten equations for these families of metrics have no irreducible solutions, so the differentials in the spectral sequence can be computed from counting only the reducible solutions. by Sherry Gong. Ph. D. 2018-09-17T15:47:38Z 2018-09-17T15:47:38Z 2018 2018 Thesis http://hdl.handle.net/1721.1/117864 1051189959 eng MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582 108 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Gong, Sherry, Ph. D. Massachusetts Institute of Technology Results on spectral sequences for monopole and singular instanton Floer homologies |
| title | Results on spectral sequences for monopole and singular instanton Floer homologies |
| title_full | Results on spectral sequences for monopole and singular instanton Floer homologies |
| title_fullStr | Results on spectral sequences for monopole and singular instanton Floer homologies |
| title_full_unstemmed | Results on spectral sequences for monopole and singular instanton Floer homologies |
| title_short | Results on spectral sequences for monopole and singular instanton Floer homologies |
| title_sort | results on spectral sequences for monopole and singular instanton floer homologies |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/117864 |
| work_keys_str_mv | AT gongsherryphdmassachusettsinstituteoftechnology resultsonspectralsequencesformonopoleandsingularinstantonfloerhomologies |