315 problems

A homotopy 4-sphere is a smooth 4-manifold homotopy equivalent to the 4-sphere $S^4$. Smooth 4-dimensional Poincaré conjecture. Every homotopy 4-sphere is diffeomorphic to $S^4$. T…

A balanced presentation of a group is a finite presentation with the same number of generators and relators. Generalized Andrews–Curtis conjecture. Any balanced presentation of the…

Smooth Poincaré conjecture in dimension 4. There are no exotic $S^4$'s; equivalently,

Let $E(n)$ be the simply connected elliptic surface, and let $E(1)_{2,3}$ denote the result of applying $2$- and $3$-log transforms to $E(1)$; this is an exotic…

Let $E(n)$ be the simply connected elliptic surface, and let $E(n)_{p,q}$ be the complex surface obtained from $E(n)$ by a pair of $p$- and $q$-log transforms, where $n\geq 1$,…

Let $d$ be an integer, and let $C$ be an algebraic curve of degree $d$ in $\mathbb{C}P^2$. Its genus is $\frac{(d-1)(d-2)}{2}$. Thom's conjecture. The genus of $C$ is minimal among…

Let $M^n_1$ and $M^n_2$ be two closed orientable $n$-manifolds. The regular genus additivity conjecture. … Regular genus is known to be subadditive under connected sum, and the equ…

Stipsicz's conjecture. Every fiber-sum indecomposable Lefschetz fibration admits an exceptional section. Equivalently, every Lefschetz fibration should be a fiber-sum of blown-up p…

A trisection of a closed 4-manifold is a decomposition into three 4-dimensional 1-handlebodies whose pairwise intersections are 3-dimensional handlebodies and whose triple intersec…

Kinoshita-type conjecture. Every $P^2$-knot in $S^4$ is of Kinoshita type.

Meier's conjecture. If $X$ admits a genus-three trisection, then $X$ is diffeomorphic to a spun lens space $S_p$ or its sibling $S_p'$, $S^4$, or a connected sum of copies of…

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Let $X$ and $Y$ be homeomorphic smooth symplectic $4$-manifolds. Equip $X\times S^2$ and $Y\times S^2$ with their product symplectic structures, and call these structures deformati…

In the Artin spun case, let knots be called essentially different when they are not isotopic, are not mirrors of one another, and are not isotopic under mirroring of connect-summan…

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphe…

Let $g\ge 1$. A handlebody of genus $g$ is a compact 3-manifold homeomorphic to a regular neighborhood of a finite connected graph with first Betti number $g$, embedded in $S^4$. B…

Let $X$ be a closed, oriented, aspherical $4$-manifold, and let $\tau(X)$ denote its signature. Gromov–Lück inequality. One has … This inequality is a refined form of the Hopf prob…

Let $S$ be a closed nonorientable surface smoothly embedded in $\mathbb{S}^4$, and let $\chi$ be its Euler characteristic. Whitney's conjecture. The Euler number of $S$ is bounded…

Let $K$ be a torus knot, and let $\gamma_4(K)$ denote its smooth 4-genus. A realizable pair $(e,h)$ records the normal Euler number and first Betti number of a smooth surface in…

Trisection conjecture for $S^4$. Every trisection of $S^4$ is isotopic to a stabilization of the genus 0 trisection.

Method-failure conjecture. The method also fails for all $h\geq 8$.

A smooth 4-manifold is said to admit a handle decomposition without 1- or 3-handles if it has a handle decomposition containing no handles of index 1 or 3. Handle decomposition con…

Let $X$ be a smooth spin rational homology $S^1\times S^3$ which is oriented and homology oriented, and let $-X$ denote $X$ with reversed orientation but the same homology orientat…

Let $M$ be a compact $4$-manifold, let $J$ be an almost complex structure on $M$, and let $h^-_J$ denote the dimension of the real vector space of closed $J$-anti-invariant $2$-for…

Let $N$ be a rational homology sphere that is the link of an isolated complex singularity, and let $q_{min}$ denote the intersection form of its minimal resolution. For any negativ…