36 problems

Honda's conjecture. These contact structures are not symplectically semi-fillable, and hence are not symplectically fillable.

Let $(Y,\xi)$ be a contact $3$-manifold. A Stein filling of $(Y,\xi)$ is a Stein domain $(N,\omega)$ with boundary contact manifold $(Y,\xi)$, and write $b_1(N)$, $\chi(N)$, and…

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

Let $(X,0)$ be a rational surface singularity, and consider its resolution. Assume that every exceptional curve in the resolution has self-intersection at most $-5$, and let the ba…

Let $D$ be a circular spherical divisor, meaning a topological divisor consisting of a cycle of spheres, and let $b^+(D)$ denote the positive index of its intersection form. The as…

Stipsicz's conjecture. Every closed contact $3$-manifold admits a universal bound on the signatures and Euler characteristics of its possible Stein fillings.

Let $(X,0)$ be a cyclic quotient surface singularity of type $\frac{1}{n}(1,q)$ with $(n,q)=1$, let $L(n,q)$ be its link, and let $\mathcal{Z}_e\left(\frac{n}{n-q}\right)$ be Lisca…

Intersection-lattice conjecture. The conclusion of Corollary still holds, under the same assumptions, if $X$ is allowed to range over the set of all symplectic fillings of $Y$.

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. The torsion of $(Y,\xi)$, denoted by … is the supremum of the integers $n\geq 1$ for which there exists a con…

Let $p>q\geq 1$ be coprime integers, let $L(p,q)$ be the corresponding lens space with contact structure $\overline\xi_{\rm st}$, and let $\mathbf Z_{p,q}$ denote the set indexing…

Nonfillability conjecture. No contact structure defined by Figure and satisfying conditions is fillable.

Multi-node filling conjecture. $\mathbb{Q}$-homology disk symplectic fillings cannot exist for resolution graphs with two or more nodes.

Let $Y=Y(e_0;r_1,r_2,\ldots,r_n)$ be a Seifert fibered space that is also a rational homology sphere, and let $\xi$ be a contact structure on $Y$. Seifert filling classification co…

Let $Y$ be a small Seifert fibered space with complementary legs and $e_0\geq 0$, equipped with a contact structure $\xi$ that is not balanced. The relevant contact structures are…

Boundary Dehn twist conjecture. The boundary Dehn twist $\tau_M$ has infinite order in the smooth mapping class group

Uniqueness conjecture. The Stein structures on these fillings are unique up to Stein homotopy.

The $-D(p,q)$ filling conjecture. The manifold $-D(p,q)$ admits a symplectic rational homology ball filling if and only if the filling is constructed from the cited cable construct…

The $\mathcal{QHB}$ conjecture. If $Y$ admits a symplectic rational homology ball filling, then

Let $\xi_n$ be the contact structure on $\mathbb{RP}^{2n+1}$ induced by a connection $1$-form on the Boothby--Wang circle bundle over…

A rational singularity is a normal surface singularity whose resolution has geometric genus zero, and an L-space is a rational homology 3-sphere with minimal Heegaard Floer homolog…

A planar contact 3-manifold is a contact 3-manifold supported by an open book whose pages have genus zero. A minimal symplectic filling is a symplectic filling containing no symple…

Ghiggini–Van Horn-Morris non-fillability conjecture. All the other rows of tight contact structures are not exactly fillable either.

Let $W_{p,q}(\mathbf{n})$ denote the minimal symplectic filling of $(L(p,q),\xi_{can})$ corresponding to $\mathbf{n}=(n_1,\ldots,n_k)$ in the parameterizing set…

Let $\mathbb{C}^n/G$ be an isolated terminal singularity, and let $(S^{2n-1}/G,\xi_{\rm std})$ be its contact link. Terminal quotient-link orbifold-filling conjecture. Exact orbifo…

Let $\mathbb{C}^n/G$ be an isolated terminal singularity. Its contact link is $(S^{2n-1}/G,\xi_{\rm std})$. Terminal quotient-link conjecture. The contact link…