20 problems

Let $M$ be a closed orientable irreducible atoroidal 3-manifold with positive first Betti number. An integral class is a class $a\in H^2(M;\mathbb{R})$ arising from integral cohomo…

A foliated manifold $(M,\mathcal{F})$ is taut if $M$ admits a Riemannian metric for which every leaf of $\mathcal{F}$ is a minimal submanifold. The foliation is transversely orient…

Let $M$ be a compact orientable irreducible 3-manifold, let $S$ be a norm-minimising surface, and suppose a fully marked surface theorem produces a taut foliation after allowing ch…

Let $Y$ be an irreducible rational homology $3$-sphere. A group is left-orderable if it admits a strict total order invariant under left multiplication, and a co-oriented taut foli…

Let $M$ be an atoroidal $3$-manifold and let $\mathcal{F}$ be a taut foliation on $M$. Transverse pseudo-Anosov flow conjecture. The foliation $\mathcal{F}$ admits an almost-transv…

Let $M$ be a closed hyperbolic $3$-manifold whose first Betti number is at least one, and let $a \in H^2(M;\mathbb{R})$ be an even integral second cohomology class of dual Thurston…

Yazdi's virtual Euler class one conjecture. For any even integral lattice point $w\in H^2(M;{\mathbb R})$ of dual Thurston norm $1$, there exists some finite cover $\tilde{M}$ of…

Let $K$ be a knot in $S^3$ with Seifert genus $g>0$. Let $\mathcal{S}_K\subset\mathbb{Q}$ be the set of rational slopes $s$ for which there exists a taut $\mathcal{C}^0$-foliation…

Let $Y$ be an irreducible rational homology 3-sphere. A non-L-space is a 3-manifold whose Heegaard Floer homology is not "small"; a group is left-orderable when it admits a total o…

Let $K$ be a non-trivial positive braid knot, and let $g(K)$ denote its genus. For a rational number $r$, let the $r$-framed Dehn surgery along $K$ be the resulting 3-manifold. Pos…

CTF meridional detection conjecture. The meridional slope of $K$ is CTF-detected.

L-space conjecture. The following statements are equivalent:

Let $M$ be the compact $3$-manifold, let $\mathcal F$ be a taut foliation of $M$, and let $S$ be a fully marked surface, meaning that its Euler characteristic is given by pairing w…

Let $K\subset S^3$ be a knot, and let $S^3(K)$ denote its knot complement. Consider the irreducible homomorphisms … that map a chosen meridian $m$ to $\mathbf{i}\in SU(2)$. Assume…

Juhász's conjecture. The sutured manifold $(M,\gamma)$ admits a taut foliation of depth at most $2k$.

A composite knot is a nontrivial connected sum of knots, and a knot is persistently foliar if, for each boundary slope, its complement admits a co-oriented taut foliation meeting t…

Let $K$ be a knot in $S^3$ that is not an L-space knot, meaning that no nontrivial Dehn surgery on $K$ yields an L-space. A persistently foliar branched surface in the exterior of…

A taut foliation on a $3$-manifold is a codimension-one foliation for which every leaf meets a transverse closed curve. An L-space is a rational homology $3$-sphere whose Heegaard…

Let $k$ be a nontrivial knot in $S^3$, and let $M=S^3\setminus\operatorname{int}(N(k))$ be its exterior. For slopes $s\in(-a,b)$, require that $M$ have a taut foliation whose restr…

A foliated manifold is uniform if its leaf space is compact. A Riemannian foliation is complete when it admits a complete bundle-like Riemannian metric, and it is strongly tense wh…