119 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…

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

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…

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 $K$ and $K'$ be knots, and let $S^3_0(K)$ and $S^3_0(K')$ denote their 0-surgeries. Akbulut–Kirby conjecture. If $K$ and $K'$ have homeomorphic 0-surgeries, then $K$ and $K'$ a…

Let $c_b(a)$ be the infimum of the $\lambda>0$ for which the ellipsoid $E(1,a)$ symplectically embeds into the polydisc $P(\lambda,\lambda b)$, and let … … The associated exceptiona…

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…

Let $X_1$ and $X_2$ be smooth, spin, closed, oriented four-manifolds, and let … be their generalized fiber sum, also smooth, spin, closed, and oriented. Let $d_1$ and $d_2$ be thei…

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…

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…

BLF achirality conjecture. Not all closed, smooth, oriented $4$-manifolds admit BLFs; in particular, it is possible that $\mathbb{C}P^2$ is necessarily achiral as a fibration, even…

Infinite-value conjecture. There is a suitable homeomorphism type on which Perelman's invariant takes on infinitely many distinct values.

Let $M$ be a smooth orientable four-manifold that admits a proper geometric decomposition into pieces modelled on ${\mathbb H}^2 \times {\mathbb H}^2$, and let ${\rm h}(M)$ denote…

Let $X$ be a closed oriented even $4$-manifold, meaning that its intersection form is even, and let $\sigma(X)$ denote its signature. The $\frac{5}{4}$-conjecture. One has … This i…

Let $X$ be a closed oriented $4$-manifold with $b_2^+(X)=1$. The period map sends a Riemannian metric to the self-dual line $H^+$ in the cone $\mathcal{P}$ of elements of positive…

Let $Y$ be a rational homology sphere with $|H_1(Y;\mathbb Z)|=h$. Suppose that $Y$ bounds a negative-definite four-manifold $X$ whose first homology has no torsion. Write…

Let $(M,\omega)$ be a symplectic four-manifold containing a top-dimensional submanifold with boundary that is diffeomorphic to an almost toric manifold whose base has a one-stratum…

Let $M^n$ be a connected closed manifold whose fundamental group $\pi_1(M)$ is finite of odd order. A metric of positive scalar curvature on the universal cover of $M$ need not des…

Hyperwinding family conjecture. $M$ is diffeomorphic to the underlying smooth manifold of a $K3$ surface, and the hyperwinding family of symplectic forms is homotopic, through…

Let $M^4$ be a four-manifold with a taut foliation ${\mathcal F}$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surface $S$ in…

Let $M^4$ be a $4$-manifold, let ${\mathcal F}$ be a taut foliation on $M$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surfa…

Pu–Loewner inequality conjecture. Every such metric satisfies

Universal stable 2-systolic inequality conjecture. There is a numerical constant $C$, independent of $X$, such that for every metric $\mathbf{g}$ on $X$,

Symplectic circle-action conjecture. Every symplectic 4-manifold which admits a circle action also admits (possibly a different) symplectic form and circle action which are symplec…

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…