Composition and Cobordism Maps

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dc.contributor.advisorLipshitz, Robert
dc.contributor.authorCohen, Jesse
dc.date.accessioned2024-01-09T21:07:26Z
dc.date.available2024-01-09T21:07:26Z
dc.date.issued2024-01-09
dc.description.abstractWe study the relationship between the algebra of module homomorphisms under composition and 4-dimensional cobordisms in the context of bordered Heegaard Floer homology. In particular, we prove that composition of module homomorphisms of type-$D$ structures induces the pair of pants cobordism map on Heegaard Floer homology in the morphism spaces formulation of the latter, due to Lipshitz--Ozsv\'{a}th--Thurston. Along the way, we prove a gluing result for cornered 4-manifolds constructed from bordered Heegaard triples. As applications, we present a new algorithm for computing arbitrary cobordism maps on Heegaard Floer homology and construct new nontrivial $A_\infty$-deformations of Khovanov's arc algebras. Motivated by this last result and a K\"{u}nneth theorem for Heegaard Floer complexes of connected sums, we also prove the existence of a tensor product decomposition for arc algebras in characteristic 2 and show that there cannot be such a splitting over $\Z$.en_US
dc.identifier.urihttps://hdl.handle.net/1794/29075
dc.language.isoen_US
dc.publisherUniversity of Oregon
dc.rightsAll Rights Reserved.
dc.subject3-manifold invariantsen_US
dc.subjectcobordismen_US
dc.subjectdeformation theoryen_US
dc.subjectFloer homologyen_US
dc.subjectKhovanov homologyen_US
dc.subjecttopologyen_US
dc.titleComposition and Cobordism Maps
dc.typeElectronic Thesis or Dissertation
thesis.degree.disciplineDepartment of Mathematics
thesis.degree.grantorUniversity of Oregon
thesis.degree.leveldoctoral
thesis.degree.namePh.D.

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