119 problems

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…

Let $M^{4}=\overline{{\mathbb C}{\mathcal H}_{2}/\Gamma}$ be a reverse-oriented compact complex-hyperbolic $4$-manifold. Reverse-oriented complex-hyperbolic vanishing conjecture. A…

Let $(X,\omega_X)$ and $(Y,\omega_Y)$ be symplectic four-manifolds that are homeomorphic. Write $\omega_{st}$ for the standard Kähler form on projective space, and identify their s…

Let $(X1, \Sigma1)$ and $(X2,\Sigma2)$ be permissible 4-manifolds with $b1(Xi)=0$, and let $tilde Xi=Xi{Sigmai}Bg$. Write … Choose an identification $\phi$, set $X=X_1\#_{\Sigma}X_2$…

Let $X$ be a manifold of simple type. For a choice of $Spin_c$-structure $L$, let $M_L$ be the Seiberg–Witten moduli space and let $d=\dim(M_L)$. Simple-type conjecture. The only n…

Let $X$ be a smooth, compact, oriented, simply connected $4$-manifold without boundary. A connected sum here means a decomposition into smooth four-manifolds, and both orientations…

A smooth, compact, oriented $4$-manifold without boundary is understood as a connected sum of symplectic manifolds, allowing either the symplectic or the opposite orientation, and…

A smooth, closed and oriented $4$-manifold is a smooth manifold of dimension four with the stated properties. A symplectic manifold is a smooth manifold equipped with a symplectic…