Results on spectral sequences for monopole and singular instanton Floer homologies

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2018

Author dc:creator

• Gong, Sherry, Ph. D. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Tomasz S. Mrowka.

• Mathematics.

Statement dc:rights

• 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.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/117864

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/117864