arXiv subjects

Jesse Cohen

Publications and source records attributed to Jesse Cohen.

4 recordsLinked to original sources

arXiv2025

Immersed Curves and 4-Manifold Invariants

For 3-manifolds with torus boundary, the bordered Heegaard Floer invariants of Lipshitz--Ozsv\'ath--Thurston have a geometric interpretation as immersed multi-curves with local systems in the punctured torus according to the work of Hanselman--Rasmussen--Watson. We consider morphisms between these immersed curve invariants and show that they compute certain cobordism maps. More precisely, we relate composition in the Fukaya category of immersed curves in the punctured torus to composition of morphisms between the bordered Floer invariants, which have interpretations in terms of certain cobordism maps. We make use of this formalism to obstruct smooth equivalences between 4-manifolds with boundary, and between surfaces with boundary in the 4-ball.

math.GT↗

arXiv2024

An Ozsv\'{a}th--Szab\'{o}-type spectral sequence for links in $S^1\times S^2$

We show that there is a spectral sequence with $E^2$-page given by the Khovanov homology of a link in $S^1\times S^2$, as defined by Rozansky in arXiv:1011.1958, which converges to the Hochschild homology of an $A_\infty$-bimodule defined in terms of bordered Floer invariants. We also show that the homology algebras $H_*\mathfrak{h}_n$ of the algebras $\mathfrak{h}_n$ over which these bimodules are defined give nontrivial $A_\infty$-deformations of Khovanov's arc algebras $H_n$ for $n>1$.

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arXiv2023

Composition Maps in Heegaard Floer Homology

We use results of Auroux arXiv:1001.4323 and Zemke arXiv:1801.09270 to prove that, in the morphism spaces formulation of Heegaard Floer homology given in arXiv:1005.1248, the opposite composition map agrees up to homotopy with the map on Heegaard Floer complexes induced by a pair-of-pants cobordism. As an application, we give an algorithm for computing arbitrary cobordism maps on hat Heegaard Floer homology.

math.GT↗

arXiv2022

An Exceptional Splitting of Khovanov's Arc Algebras in Characteristic 2

We show that there is an associative algebra $\widetilde{H}_n$ such that, over a base ring $R$ of characteristic 2, Khovanov's arc algebra $H_n$ is isomorphic to the algebra $\widetilde{H}_n[x]/(x^2)$. We also show a similar result for bimodules associated to planar tangles and prove that there is no such isomorphism over $\mathbb{Z}$.

math.GT↗