240 problems

Let $K\subset S^3=\partial D^4$ be a knot. It is slice if it bounds a smoothly, properly embedded disk in $D^4$; it is ribbon if it bounds a disk whose only singularities are ribbo…

Let $X$ be a smooth four-manifold, let $\xi$ be a class defining a wall, and let $\Delta_{\xi,d}^X$ denote the corresponding wall-crossing term for Donaldson invariants. The multip…

Flatness conjecture. Then $M^4$ is flat.

Kervaire–Laudenbach conjecture. If $G$ is nontrivial, then

Let $K$ and $K'$ be knots in $S^3$. Their zero surgeries are the $3$-manifolds $S^3_0(K)$ and $S^3_0(K')$, respectively, and the knots are concordant when they cobound a smooth ann…

Let $M^4$ be a closed, oriented, connected $4$-manifold. A simple branched covering is a branched covering whose local monodromies around the branch set are transpositions; let…

A closed hyperbolic four-manifold is a closed four-dimensional manifold equipped with a hyperbolic structure. Its Seiberg--Witten invariants are the invariants associated with its…

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$ and odd $b^+(X)-b_1(X)$. A Seiberg–Witten basic class is a cohomology class associated wi…

Let $(X,\omega)$ be a symplectic $4$--manifold. A Weinstein trisection of $(X,\omega)$ is the notion proposed as an appropriate generalization of the standard trisection of…

Mariño–Moore's conjecture. There should be a generalisation of Witten's conjecture to these higher-rank invariants, implying in particular that they do not contain new differential…

Let $X$ be a simply connected four-manifold with $b_+(X)>1$. A manifold is of finite type if there is a non-negative integer $r$ such that … for the relevant Donaldson invariant…

Let $(M^{4},g)$ be a compact anti-self-dual Einstein manifold with negative scalar curvature. Anti-self-dual Einstein vanishing conjecture. All the Seiberg–Witten invariants vanish…

Let $X$ be a minimal symplectic $4$-manifold, meaning that it contains no symplectically embedded exceptional sphere, and suppose that $X$ is not diffeomorphic to a ruled surface.…

Donald's conjecture. The structure $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to the triple of Kähler forms arising from a hyperkähler met…

Let $f:X\to\mathbb{S}^2$ be a relatively minimal Lefschetz fibration, meaning that no fiber contains a $(-1)$-sphere. It is decomposable if it can be expressed as a fiber sum of tw…

Let a knot group be a group of deficiency $1$, and suppose it is an HNN extension with finitely generated base group. Silver's conjecture. The base group is finitely presentable. T…

Let $(M,\partial M)\longrightarrow (X,\partial X)$ be a degree one normal map from a topological $4$-manifold to a Poincaré pair, and suppose it is a $\mathbb{Z}[\pi_1 X]$-homology…

Nonexistence conjecture. There is no such manifold for which

Let $(M,\domega)$ be a minimal symplectic $4$-manifold with symplectic Kodaira dimension $\delta0kappa^s(M)=2$. Bogomolov–Miyaoka–Yau conjecture. The symplectic canonical class sat…

Let $M$ be a symplectic $4$-manifold with canonical class $K=0$, and let $vb_1(M,\Bbb F_p)$ denote its virtual first Betti number with coefficients in the finite field $\Bbb F_p$.…

The compact-case conjecture. Any compact, 4-dimensional almost Kähler manifold of pointwise constant totally-real sectional curvature is a self-dual Kähler surface.

Let $E(2)$ be the elliptic K3 surface, and let $X_{K_1}$ and $X_{K_2}$ denote the manifolds obtained from $E(2)$ by knot surgery using knots $K_1$ and $K_2$, respectively. Knot-sur…

Finiteness and uniqueness conjectures. The class $\alpha$ should be represented by at most finitely many connected embedded symplectic submanifolds; the stronger conjecture is that…

Let $X$ be a closed, oriented, smooth four-manifold with $b_2^+(X)>0$. Let ${\mathfrak{s}}={\mathfrak{s}}_0\otimes L_1$ be a $\operatorname{Spin}^c$ structure defined by…

Kotschick–Morgan's weak conjecture. The transition formula $\delta_{X,{\mathcal C}_+,{\mathcal C}_-}^w$ is a homotopy invariant of the pair $(X,\xi)$; more precisely, if $\phi$ is…