17 problems

Let a Brieskorn homology sphere be a 3-dimensional link $\Sigma(a_1,\ldots,a_n)$ of an isolated complete intersection singularity as described above, and call it nontrivial when it…

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…

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

Let $\xi_0$ be the unique tight contact structure on $-\Sigma(2,3,11)$, and let $N(2)$ denote the nucleus referred to in the conjecture. Nucleus conjecture. Any Stein filling of…

Let $\Gamma_g^r$ be the mapping class group of a genus-$g$ surface with $r>0$ boundary components. A right-handed Dehn twist is a Dehn twist with the positive orientation conventio…

Finiteness conjecture. The set $\mathcal{C}_{(M,\xi)}$ is finite.

Let $S$ be a Stein filling of the 3-torus $T^3$. The preceding construction shows that the closed spin 4-manifold obtained by gluing $S$ to $Y\mathbin{\setminus}\nu F$, where $Y$ i…

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

Let $(W,J)$ be the rational homology ball Stein filling of $L(9,2)$, and let $N_k\subset L(9,2)$ be the link shown in the paper's figure. The links $N_{3d}$ are considered for inte…

The fillings under consideration are the explicit Stein fillings associated with the rational cuspidal curves of types $\mathcal{A}_p$ and $\mathcal{B}_p$, and Figures … give diagr…

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…

Let a cusp singularity be a surface singularity whose link carries the contact structure associated with its cusp resolution, and let a rational dual mean that the dual cusp singul…

Let $Y$ be an integer homology $3$-sphere bounding a Stein $4$-manifold $(X,J)$ with $c_1(J)\neq 0$. Stein representation conjecture. There exists a nontrivial homomorphism … The c…

Let $X$ be a smooth $4$-manifold with Stein structures $J_1$ and $J_2$ such that $c_1(J_1)-c_1(J_2)$ is nontorsion. Let $Y=\partial X$, and let $\xi_1$ and $\xi_2$ be the induced c…

Ozbagci–Stipsicz conjecture. The set $\mathcal{C}_{(M, \xi)}$ is finite.

Stein fillability conjecture. A homotopy sphere $\Sigma^{2q+1}$ is Stein fillable if and only if

Non-Stein-fillability conjecture. The contact structures $\eta_{i,j}^n$ are not Stein fillable if $i>0$.