Introduction

The purpose of this site is to record forward-looking problems of interest around Floer homology as used in low-dimensional topology. The focus of this list is meant to be narrower than that of the Kirby list, the problems more approachable.

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Bordered HF and Fox-Milnor for slice fibered knots

Problem: Categorify the Fox-Milnor criterion for slice fibered knots. If $K\subset S^3$ is a slice knot (either topologically or smoothly) then its Alexander polynomial satisfies $$\Delta_K(t) = f(t)f(t^{-1})$$ for some $f\in \mathbb{Z}[t,t^{-1}]$. This is called the Fox-Milnor criterion . Knot Floer homology is said to categorify  the Alexander polynomial; more precisely, its graded Euler characteristic satisfies $$\chi(\widehat{\mathit{HFK}}(K)) = \bigoplus_{m,a}(-1)^m t^a \dim \widehat{\mathit{HFK}}(K)  = \Delta_K(t).$$ It is therefore natural to ask (the admittedly vague question of) whether there are extra features of the knot Floer homology of slice knots which recover the Fox-Milnor criterion for the Alexander polynomial. One would hope moreover that these features provide a stronger obstruction to slicing a knot than the classical Fox-Milnor criterion. This is one thing that could be meant by the charge to categorify  Fox-Milnor.  It's not at all clear howev...
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The 4-holed sphere and tight contact structures

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Problem: Classify open books with page the 4-holed sphere which correspond to tight contact structures. Recall that an open book is a pair $(S,h)$ where $S$ is a compact, oriented surface with boundary, and $h$ is a diffeomorphism of $S$ which restricts to the identity on $\partial S$. We will refer to $h$ as the monodromy  of the open book and $S$ as the page. A construction of Thurston-Winklenkemper assigns a contact 3-manifold $(M_{S,h},\xi_{S,h})$ to this open book, and Giroux proved that this assignment defines a bijective correspondence between open books up to positive stabilization and contact 3-manifolds up to contactomorphism (see here and here for recent proofs of Giroux's result). This problem is concerned with understanding how topological features of the monodromy are reflected in geometric features of the contact structure in Giroux's correspondence. For instance, Giroux proved that a contact 3-manifold is Stein fillable if and only if there is a correspondin...
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L-spaces and F-summands

Problem:  Prove that if $S^3_r(K)$ is not an L-space then its Heegaard Floer homology has a direct $\mathbb{F}$-summand. The L-space conjecture predicts that if an irreducible closed 3-manifold $Y$ is not an L-space then $Y$ admits a coorientable taut foliation. Recently, Lin proved in a short note that if $Y$ admits a coorientable taut foliation then its Heegaard Floer homology $\mathit{HF}^+(Y)$ has a direct $\mathbb{F}$-summand.  For background: recall that $\mathit{HF}^+(Y)$ is a module over $\mathbb{F}[U]$ where $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$. A direct $\mathbb{F}$-summand of $\mathit{HF}^+(Y)$ refers to a direct summand isomorphic as an $\mathbb{F}[U]$-module to $\mathbb{F}[U]/U$. Lin's result therefore gives a potential way to disprove the L-space conjecture: find an irreducible non-L-space whose Heegaard Floer homology does not have a direct $\mathbb{F}$-summand. Alternatively, one could try to prove that Lin's result cannot possibly give a counterexample by sh...
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