148 problems

Let $Y_\Gamma$ be a negative-definite plumbed rational homology $3$-sphere with $N$ plumbing vertices, and let $\mathfrak{s}_{\rm can}$ denote its canonical ${\rm Spin}^c$ structur…

Let $(M,g)$ be a closed orientable Riemannian three-manifold, with $g$ a metric of constant sectional curvature $\pm1$. Let $\operatorname{Isom}(M)$ be the isometry group of $M$ an…

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

Dehn-surgery conjecture. The manifold at infinity of the even subgroup of $\Delta_{3,4,\infty;n}$ is the hyperbolic $3$-manifold obtained by Dehn surgery on the first cusp of…

Let $M$ be a compact $3$-manifold admitting a metric with negative sectional curvature, and start the normalized positive cross curvature flow at such a metric. Chow–Hamilton conje…

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold containing a closed minimal surface homeomorphic to a fixed closed oriented surface of genus at least two,…

Let $M^3$ be a closed three-dimensional manifold. Yau's conjecture. The manifold $M^3$ possesses infinitely many immersed, geometrically distinct minimal surfaces. The conjecture c…

Ozsváth–Szabó's degeneration conjecture. All subsequent differentials in the universal coefficient spectral sequence vanish, so that its $E^\infty$ term is $HC^\infty_*(Y)$.

A virtual three-manifold is a Kirby equivalence class of framed virtual links, and its fundamental group is the combinatorial fundamental group associated with the virtual link. Vi…

Let $(M^3,g)$ be a complete, non-compact Riemannian manifold. Say that $(M^3,g)$ is Ricci pinched when it satisfies the Ricci-pinching condition used in Hamilton's conjecture. Hami…

White's conjecture. The number of embedded minimal tori in $(S^3,g)$ is at least $9$.

Yau's conjecture. Every Riemannian $3$-sphere contains at least $4$ embedded minimal $2$-spheres.

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the…

Rubinstein's conjecture. $H$ is either isotopic to a minimal surface or isotopic to the oriented double cover of a non-orientable minimal surface with a vertical handle attached.

Let a closed graph manifold be decomposed along incompressible tori into Seifert fibered JSJ components. Properties $(1a)$, $(1b)$, and $(2)$ denote, respectively, the existence of…

Let $\frac{r_*}{s_*}:=(e_0=\frac{r_0}{s_0},\frac{r_1}{s_1},\ldots,\frac{r_n}{s_n})$ be an $n+1$-tuple, let $J\subset\{1,\ldots,n\}$, and let $J$-realizability mean the existence of…

Lutz's conjecture. The only manifolds with $\beta_1(M,\mathbb{F})=3$ admitting a $13$-vertex triangulation are $(S^2\times S^1)^{\# 3}$ and $(S^2\uplus S^1)^{\# 3}$.

Ozsváth–Szabó's differential conjecture. The only possibly nontrivial differential in this spectral sequence is $d^3$, and this differential is given by a formula in terms of the t…

A $5/6^*$-triangulation of a $3$-manifold is a triangulation in which every edge has degree $5$ or $6$, meaning that $5$ or $6$ tetrahedra meet around each edge, and every triangle…

Let $(M,g(t))$ be a solution to the Ricci flow on a compact $3$-manifold with arbitrary initial metric. Hamilton's Harnack-estimate conjecture. A Harnack-type estimate holds for th…

Let a finite-time singularity of the Ricci flow on a compact $3$-manifold be dilated, and suppose a subsequential limit is a quotient by isometries of one of $S^3$,…

The conjecture. For every nontrivial knot $K$,

Let $K(a_0,a_1,a_2)$ be the link of the surface singularity … at the origin, and set $s=1/a_0+1/a_1+1/a_2$. Milnor's conjecture. If $s\leq 1$, then the fundamental group of the cor…

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitt…

Torsion Euler class conjecture. The Euler class $e(cxi)$ is torsion.