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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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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Knot Floer homology and twist coefficients

Question: What information does the knot Floer complex of a fibered knot record about the fractional Dehn twist coefficient of its monodromy? Let $\phi$ be a diffeomorphism of a surface $S$ with connected boundary, which restricts to the identity on the boundary. The fractional Dehn twist coefficient of $\phi$, denoted by $c(\phi)$, is a measure of the twisting near the boundary of the surface in the free isotopy between $\phi$ and its Nielsen--Thurston representative. It was defined by Honda-Kazez-Matic as a quantification of their notion of right-veering surface diffeomorphisms, which in turn came from their study of contact structures on 3-manifolds (a contact structure is tight if and only if all of its supporting open books have right-veering monodromy). It is also closely related to and inspired by Gabai's notion of degeneracy slope . Suppose for example that $\phi$ is freely-isotopic to a pseudo-Anosov map $\phi_0$. The latter fixes a pair of transverse, singular, tran...
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Left-orderability and SU(2)-representations

Problem: Prove that if $Y$ is an irreducible 3-manifold with left-orderable fundamental group then there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image. Suppose $Y$ is irreducible with left-orderable fundamental group. Then the L-space conjecture predicts that $Y$ is not an L-space. Together with Kronheimer-Mrowka's conjectured isomorphism $$\widehat{\mathit{HF}}(Y;\mathbb{C}) \cong_{\mathrm{Conj.}} I^\sharp(Y;\mathbb{C})$$ between Heegaard Floer homology and framed instanton homology, this would imply that $Y$ is not an instanton L-space. If $Y$ satisfies a further mild non-degeneracy condition (and perhaps even if $Y$ doesn't) then being an instanton non-L-space implies that there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image, per  Baldwin-Sivek . While these conjectures serve as motivation, one can try to prove the stated link between left-orderability and nonabelian $SU(2)$-representations directly. That's the goal of this problem...
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