458 problems

Borel conjecture. Every homotopy equivalence

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…

Poincaré conjecture. Every closed simply connected $3$-manifold is homeomorphic to the $3$-sphere $S^3$. This conjecture was proved by Perelman as part of the resolution of the geo…

Let $\Gamma$ be a discrete group and consider the assembly map from the $K$-homology of an appropriate classifying space of $\Gamma$ to the $K$-theory of the reduced $C^*$-algebra…

A closed aspherical manifold is a closed manifold whose universal cover is contractible. Gromov–Lawson conjecture. A closed aspherical manifold cannot support a Riemannian metric o…

A hyperbolic $3$-manifold is algebraically finite when its fundamental group is finitely generated, and it is topologically finite when it is homeomorphic to the interior of a comp…

Let the space of embedded two-spheres in $\mathbb{R}^3$ be the space of embeddings of $S^2$ into $\mathbb{R}^3$, with its natural topology. Smale's conjecture. In one of its many f…

Bing–Borsuk conjecture. Every $n$-dimensional homogeneous metric ANR-compactum is an $n$-manifold.

Let $K$ be a contractible $2$-complex, and let $I$ be the closed interval. Zeeman's conjecture. The product $K\times I$ can be collapsed to a point. This conjecture is a classical…

Stable Andrews–Curtis conjecture. Stable Andrews–Curtis moves should suffice to reduce every balanced presentation of the trivial group to the standard one-generator presentation.

A $3$-manifold is understood in the context of low-dimensional topology, and the Poincaré conjecture asserts that a $3$-manifold homotopy equivalent to the $3$-sphere is homeomorph…

Busemann's conjecture. Every G-space is finite-dimensional, and every finite-dimensional G-space is a topological manifold.

Gromov's conjecture. $\mathbb{Z}_2$-systolic freedom does not exist.

Virtual Haken conjecture. Every compact, orientable, irreducible 3-manifold with infinite fundamental group is virtually Haken.

Unstable Gromov–Lawson–Rosenberg conjecture. The vanishing of $\alpha(M)$ decides whether $M$ admits a metric of positive scalar curvature.

Halpern–Weaver conjecture. The infimal aspect ratio of an embedded paper Möbius band is

A group is a $PD_n$ group if its fundamental-group cohomology satisfies Poincaré duality over $\mathbb Z$ in dimension $n$. Wall Conjecture. Conversely, every $PD_n$ group is the f…

Generalized Poincaré conjecture. Any homotopy $n$-sphere is homeomorphic to $S^n$.

Four-dimensional Smale conjecture. The group $\mathrm{Diff}_{\partial}(D^4)$ is contractible.

Gromov–Lawson–Thurston conjecture. A necessary condition for the disc bundle to admit a hyperbolic structure is

Let $M$ be a compact, connected, $P^2$-irreducible $3$-manifold. Say that $M$ has almost-compact coverings if, for every finitely generated subgroup $H$ of $\pi_1(M)$, the cover of…

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 $W_0$ and $W_1$ be closed, connected, oriented $3$-manifolds. A rational homology ribbon cobordism from $W_0$ to $W_1$ is a ribbon cobordism $W$ from $W_0$ to $W_1$ such that t…

Let $M$ be a manifold of dimension $m$, and let $T^m$ denote the $m$-dimensional torus. Generalized Geroch conjecture. There is no complete metric of positive scalar curvature on ……

Let $\mathcal{M}_n$ be the class of closed oriented $n$-manifolds, and let $\gtrsim$ denote domination by a continuous map of nonzero degree. Let…