21 problems

Let $M$ be a closed $3$-manifold. Thurston's geometrization conjecture. Every closed $3$-manifold can be decomposed into pieces, each admitting one of eight geometric structures. T…

Let $M$ be a compact oriented $3$-manifold with finite fundamental group. A spherical $3$-manifold is a quotient $\mathbb{S}^3/G$, where $G$ is a finite subgroup of…

Hyperbolization Conjecture. $M$ is hyperbolic.

Let $M$ be a closed, orientable, atoroidal 3-manifold, meaning that it contains no essential torus. Thurston's hyperbolization conjecture. If $M$ has infinite fundamental group, th…

Geometrisation Conjecture. Every compact orientable $3$-manifold has a decomposition into geometric pieces.

Let $G$ be a finite group, and write $G\in\mathcal{G}^3$ when there is a closed $3$-manifold $M$ with $\pi_1(M)\cong G$. Finite fundamental group conjecture. The only finite groups…

Let $M$ be a closed, oriented 3-manifold. A manifold is tame when it satisfies the paper's tameness condition, and spherically tame when it satisfies the corresponding spherical co…

Let $M$ be a closed, oriented, irreducible 3-manifold with $\sigma(M)<0$. A minimizing sequence for $\mathcal{S}_{-}^{2}$ is a sequence of metrics approaching…

Let $M$ be a closed, irreducible $3$-manifold with infinite fundamental group, and suppose that $e53_1(M)$ contains no subgroup isomorphic to $e53\opluse53$. A closed $3$-manifold…

Let $M$ be a closed 3-manifold. A closed 3-manifold is simple if it contains no essential surface of non-negative Euler characteristic, such as a sphere, disk, annulus, or torus. G…

The covering conjecture for closed irreducible 3-manifolds. The manifold $M$ is covered by $\mathbb R^3$.

Geometrization conjecture. Every $3$-manifold is geometrically modeled by one of these eight Thurston geometries. The conjecture was proved by G. Perelman in his work toward the pr…

Let $M$ be a compact, orientable, prime $3$-manifold. The eight Thurston geometries are the homogeneous geometries that admit compact quotient manifolds. Thurston's geometric decom…

Let $(M^4,J)$ be a compact Class $\mathrm{VII}_+$ surface, let $\omega_0$ be a pluriclosed metric on $M$, and let $p\in M$ be a generic point as described in the surrounding discus…

Let $(M^4,J)$ be a primary Hopf surface, and let $\omega_0$ be a pluriclosed metric on $M$. Primary Hopf surface convergence conjecture. The solution $\omega_t$ to pluriclosed flow…

Let $(M^4,J)$ be an Inoue surface, and let $\omega_0$ be a pluriclosed metric on $M$. Inoue surface geometrization conjecture. The solution $\omega_t$ to pluriclosed flow with this…

Let $(M^4,J)$ be a properly elliptic surface with $\operatorname{kod}(M)=1$ and odd first Betti number, and let $\omega$ be a pluriclosed metric. Properly elliptic geometrization c…

Geometrization conjecture. There is a finite collection $\{S_i\}$ of incompressible orientable Euclidean $2$-dimensional suborbifolds of ${\mathcal O}$ such that every connected co…

Let $M$ be a closed $3$-manifold. Geometrization conjecture. Every such $M$ admits a piecewise locally homogeneous metric. This is presented as a version of Thurston's Geometrizati…

Elliptization Conjecture. $M$ is spherical.

A closed 3-manifold is a compact 3-manifold without boundary, and it is simply connected when its fundamental group is trivial. The Poincaré conjecture asserts that every closed, s…