28 problems

Finiteness conjecture. Every planar contact $3$--manifold should admit only finitely many deformation classes of minimal symplectic fillings.

Let $S$ be a compact, oriented surface with boundary, let $\operatorname{Map}(S,\partial S)$ be the group of orientation-preserving diffeomorphisms of $S$ that restrict to the iden…

A right-veering open book is an open book whose monodromy is right-veering, and it is minimal when it is not destabilizable. Minimal right-veering open book conjecture. Every minim…

Torsion Euler class conjecture. The Euler class $e(cxi)$ is torsion.

Let $\mathcal M_k$, for $k\geq 0$, be the set of closed, oriented, smooth $3$-manifolds admitting open book decompositions whose monodromies contain at most $k$ negative Dehn twist…

In dimension $n=3$, an open book is a decomposition of a $3$-manifold into a page and monodromy; the standard open book on $S^3$ is $(D^2,\operatorname{id})$. A Hopf plumbing or de…

Arbitrarily large Weinstein $\mathcal{L}$-invariant conjecture. The Weinstein domains in Example 1 and Example 2 have arbitrarily large Weinstein $\mathcal{L}$-invariants under thi…

The open-book conjecture.

Open-book subsurface conjecture. $(Y,\xi)$ is tight if and only if every subsurface of the page of each supporting open book is Thurston norm minimizing and, moreover, has minimal…

Let $(M,\xi)$ be a contact manifold with an open book whose $(2n-2)$-dimensional page is $P$. Let $L$ be a Lagrangian $(n-1)$-disk with $\partial L\subset \partial P$, and suppose…

Planarity conjecture. No tight, positive contact structure on a Brieskorn homology sphere is supported by a planar open book decomposition.

Let $X$ be the Stein manifold obtained from $B^4$ by attaching a Stein $2$-handle along the maximal Thurston–Bennequin right-handed Legendrian torus knot in $(S^3,\xi_{std})$, and…

Harer's stabilization conjecture. Stabilizations and destabilizations are sufficient to relate any two open book decompositions of the same $3$-manifold.

Boundary P-bigon conjecture. Non-existence of a boundary $P$-bigon is equivalent to being non-quasi-right-veering.

Planar filling finiteness conjecture. Every planar contact $3$-manifold has at most finitely many distinct deformation classes of minimal symplectic fillings.

Relative Giroux correspondence. Every compact contact manifold with convex boundary is contactomorphic to

Conjecture prime. If $\delta(\mathcal{T},\alpha)=0$, so that the Bennequin–Eliashberg inequality is sharp on $(\mathcal{T},\alpha)$, then $\mathcal{T}$ is represented by an…

Minimum-genus alpha-Bennequin surface conjecture. If $\delta(\mathcal{T},\alpha)\geq 0$, then $\mathcal{T}$ bounds a minimum-genus $\alpha$-Bennequin surface with respect to…

Gabai's conjecture. If the open book is stabilized, then

Positive stabilization conjecture. The new contact manifold $(M,\xi')$ is conjectured to be contactomorphic to $(M,\xi)$.

Let $Y$ be a rational homology $3$-sphere, let $\tau$ be a contact structure on $Y$, and let $\tau$ induce a $\textrm{spin}^c$ structure on $Y$ with correction term $d$. Let…

No-torsion complement conjecture. The complement $Y\setminus B$ of the binding of an open book for any contact structure has no Giroux torsion. This is presented as a very weak for…

Let $(Y,\xi)$ be a contact 3-manifold. The binding number $bn(\xi)$ is the minimal number of binding components among open books supporting $\xi$ whose pages have minimal genus. Th…

Let $(M^{2n+1},\alpha)$ be a contact manifold with a supporting open-book decomposition. Attach an $n$-handle to the page, producing a Lagrangian $n$-sphere $S^n$, and let the Dehn…

Let $(M,\xi)$ be a closed contact three-manifold supported by a genus one open book $(S_{1,r},\phi)$, where $\phi$ is pseudo-Anosov and the associated foliations have two singulari…