211 problems

For every integer $n\geq 2$, there exists a constant $C_n>0$ such that, for every closed oriented $n$-manifold $M$, every Riemannian metric $g$ on $M$, and every $\lambda\geq 0$, t…

For every complete Riemannian manifold $(M,g)$ with nonnegative Ricci curvature, $\operatorname{Ric}_g\ge 0$, the fundamental group $\pi_1(M)$ is finitely generated.

Let $n\geq2$, and let $\depu n$ be a decorated once punctured $n$-gon. Its full arc complex is the simplicial complex whose simplices consist of collections of pairwise compatible…

Let $X$ be an aspherical closed manifold, let $\widetilde X$ be its universal cover, and let $\pi_1(X)$ act on $\widetilde X$ by deck transformations. A closed fundamental domain i…

Let $M$ be an aspherical manifold, let $\widetilde M$ be its universal cover, and let $\pi_1(M)$ act on $\widetilde M$ by deck transformations. Contractible-translate conjecture. T…

Let $(X,d)$ be a Busemann $G$-space, meaning that it is a metric space satisfying Menger convexity, finite compactness, local extendibility, and uniqueness of extension. Suppose th…

Let $c$ be a natural number, and consider irreducible simply connected $4$-manifolds of complexity $c$. Finiteness conjecture. For every natural number $c$ there are finitely many…

Let $N$ be a hyperbolic 3-manifold and let $c_1$ and $c_2$ be closed geodesics in $N$. Their complements are the manifolds obtained by removing the geodesics. Closed geodesic compl…

Universal locally free subgroup conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free.

Let $G$ be a finitely generated group of covering translations acting on a Whitehead manifold $W$, with quotient a 3-manifold $M$. A proper plane $\Pi$ in $W$ is equivariant if, fo…

Minimax volume conjecture. The minimax volume satisfies, up to a constant factor depending only on $n$,

Let $\Gamma$ be a finitely presented $n$-dimensional Poincaré duality group over $\mathbb Z$. Wall's conjecture. There exists a closed $n$-dimensional manifold $M$ which is…

For an integer $n$, a closed hyperbolic $n$-manifold is a compact manifold locally isometric to $n$-dimensional hyperbolic space, and a closed geodesic is a geodesic that is period…

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 $M$ be a hyperbolic $3$-manifold with finitely generated fundamental group. The interior of a compact manifold with boundary is a topological model for a tame hyperbolic $3$-ma…

Let $M$ be a hyperbolic $3$-manifold with finitely generated fundamental group. Its topological type consists of its underlying topology, and its end invariants encode the geometry…

A fibered three-manifold is a three-manifold admitting a submersion $p:M\to S^1$; equivalently, it is a mapping torus … for a surface $F$ and an orientation-preserving isomorphism…

Let $M$ and $N$ be closed, orientable, topological $4$-manifolds that are homotopy equivalent, have torsion-free fundamental group, and have the same Kirby–Siebenmann invariant. St…

Let $M$ and $N$ be closed, orientable, smooth $4$-manifolds that are homotopy equivalent and have torsion-free fundamental group. Stable diffeomorphism conjecture. Then $M$ and $N$…

A simplicial complex is of type $\{3,4,5\}$ if the link of every $(n-2)$-face is one of the cycles $C_3$, $C_4$, or $C_5$. Finiteness conjecture. In every fixed dimension $n$, ther…

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…

Let $X$ be a non-degenerate compact metric space, and suppose that every midset of $X$ is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. Higher-dimensional double midset conjecture.…

Co-rank conjecture.

Let $S$ be a symmetric space of non-compact type, and let an $S$-manifold be a complete Riemannian manifold locally isometric to $S$. An irreducible $S$-manifold has no nontrivial…

Let $M^d$ be a compact, closed, $d$-dimensional manifold with a piecewise Euclidean structure, and suppose that it is non-positively curved in the sense of Gromov. Let $\chi(M^d)$…