1,106 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 rational homology $3$-sphere. Its fundamental group is left-orderable if it admits a left-invariant total order, and $M$ supports a co-orientable taut foliation if it…

Let $F_n=F(x_1,\dots,x_n)$ be the free group on generators $x_1,\dots,x_n$, and let $\langle x_1,\dots,x_n\mid r_1,\dots,r_n\rangle$ be a balanced presentation, with Andrews–Curtis…

Let $K$ be a nontrivial knot, and let $K(r)$ denote the result of $r$-Dehn surgery on $K$ for $r\in\mathbb{Q}$. Property P conjecture. The manifold $K(r)$ is never simply connected…

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 $K\subset S^3$ be a lens space knot, meaning that some positive integer surgery on $K$ produces a lens space. Berge's construction gives a family of such knots. Berge conjectur…

Cosmetic surgery conjecture. The conjecture predicts that non-trivial knots admit no truly cosmetic surgeries.

Let $M$ be a closed hyperbolic $3$-manifold. Thurston's virtual fibering conjecture. The manifold $M$ has a finite cover that fibers over the circle, equivalently, that is the mapp…

Thurston's bending conjecture. The data of these two pleating loci determine the geometry of the entire manifold $M$.

Let $Y$ be a hyperbolic integral homology sphere, and let $\lambda_1^*$ denote the first positive eigenvalue of the Laplacian on coexact $1$-forms. Ozsváth–Szabó spectral-gap predi…

Kronheimer–Mrowka conjecture. Appropriate versions of instanton Floer homology and Heegaard Floer homology are isomorphic.

Suppose that … is a tower of covers of congruence arithmetic hyperbolic 3-manifolds whose injectivity radius approaches infinity. Write $H_1(M_i,\mathbb{Z})_{tors}$ for the torsion…

Convexity question. Is the function $g$ convex?

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 $K$ be a knot in $S^3$ and let $r,s e r$ be slopes in $mathbb{Q}cup{infty}$. Write $S^3r(K)$ for the manifold obtained by Dehn surgery on $K$ with slope $r$; two surgeries are…

Let $L=K_1\cup\dots\cup K_n$ be an $n$-component link in $S^3$. A 0-framed surgery on $L$ means surgery with framing zero on every component, and let $\#_n(S^2\times S^1)$ denote t…

Let a surgery on a knot in a three-manifold produce a lens space. In the context above, non-torus knots admit at most two such surgeries, and these slopes must be successive intege…

Let $M$ be a closed orientable irreducible atoroidal 3-manifold with positive first Betti number. An integral class is a class $a\in H^2(M;\mathbb{R})$ arising from integral cohomo…

Let $f$ be a transitive partially hyperbolic diffeomorphism of a $3$-manifold. A diffeomorphism is a variable time map of a flow if the time spent along each orbit is given by a su…

A knot in $S^3$ is a 2-generator knot if its knot group can be generated by two elements. A knot has tunnel number one if it admits a tunnel system consisting of one tunnel. Tunnel…

For each $g\geq 3$, let the genus-$g$ Goeritz group of $S^3$ be the group of orientation-preserving homeomorphisms of the genus-$g$ Heegaard splitting of $S^3$, modulo isotopy. Pow…

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…

Let $f$ be a partially hyperbolic diffeomorphism on a 3-manifold, with invariant splitting $TM=E^s\oplus E^c\oplus E^u$. Write $E^{cu}=E^c\oplus E^u$ and $E^{sc}=E^s\oplus E^c$. Th…

Let $f$ be a volume-preserving partially hyperbolic diffeomorphism of a $3$-manifold with invariant splitting $TM=E^s\oplus E^c\oplus E^u$. A torus is tangent to the…

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