139 problems

Goda–Teragaito lens space surgery conjecture. Then $K$ is a fibered knot and

Minimality conjecture. For $p,q2$, the triangulations $\mathcal{T}_{K(p,q)}$ realise the triangulation complexity of the complement of $K(p,q)$; equivalently, they use the minimal…

A knot $K$ is chiral if it is not equivalent to its mirror image, and $T(2,n)$ denotes the $(2,n)$ torus knot. Chirally cosmetic surgery conjecture. If $K$ is a chiral knot that is…

Let $K$ be a knot in $S^3$. A surgery on $K$ is reducible if the resulting 3-manifold is reducible, and $K$ is a cabled knot if it is obtained as a cable of another knot. The cabli…

Let $M$ be a closed, oriented $3$-manifold, and let $K_1,K_2\subset M$ be knots. Their complements are the compact $3$-manifolds obtained from $M$ by removing open tubular neighbor…

A closed 3-manifold $M$ has a Heegaard surface of fixed genus $g$. Such a surface is irreducible if it does not arise by stabilization from a lower-genus Heegaard splitting. Waldha…

Let $M_3$ be a 3-manifold and let ${\rm Tor}\,H_1(M_3;{\mathbb Z})$ denote the torsion subgroup of its first homology. A $q$-series $\hat{Z}_a(q)$ is associated to each…

Let $M=\left\{b;\left(\epsilon,g,(t,k)\right);\left(\ \mid \ \right); \left((p_1,q_1),\ldots,(p_r,q_r)\right)\right\}$ be a non-orientable closed irreducible and $\mathbb P^2$-irre…

Let $M_1$ and $M_2$ be compact orientable $3$-manifolds with incompressible boundary components $F_1$ and $F_2$, respectively, and let $h:F_1\rightarrow F_2$ be any homeomorphism.…

Let $M$ be a compact irreducible $3$-manifold, and let $N$ be a cover of $M$ with finitely generated fundamental group. Simon's conjecture. The interior of $N$ is topologically tam…

A hyperbolic Mom-$4$ manifold is a hyperbolic $3$-manifold equipped with a Mom-$4$ structure. Mom-4 conjecture. The collection of hyperbolic Mom-$4$ manifolds is exactly the set of…

Let $W$ be a genus-two handlebody embedded in a compact, orientable 3-manifold $M$, and let $\alpha$ and $\beta$ be refilling meridians of $W$. Denote the corresponding refilled ma…

Let $K_1$ and $K_2$ be knots in a closed, oriented 3-manifold $M$, and suppose that their complements are homeomorphic by an orientation-preserving homeomorphism. Oriented knot com…

Fibered-cover characterization conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free if and only if $N$ does not have a finite cover w…

Let $M$ be a compact, orientable, hyperbolizable 3-manifold with incompressible boundary. Let $\{M_1,\ldots,M_n\}$ be the components of $M-\Sigma(M)$ that are not homeomorphic to s…

Let $M$ be a compact, orientable, hyperbolizable 3-manifold, and let $V(M)$ be the infimum of the volumes of the convex cores of hyperbolic manifolds homeomorphic to the interior o…

Let $N$ be a complete hyperbolic 3-manifold with finitely generated fundamental group. A hyperbolic 3-manifold is topologically tame if it is homeomorphic to the interior of a comp…

Let $M^{3}$ be a closed, prime, atoroidal $3$-manifold. A locally homogeneous Riemannian metric is a Riemannian metric such that for any two points $p$ and $q$ there is an isometry…

The paper considers compact $3$-manifolds and canonical decompositions into geometric pieces. Geometric decomposition conjecture. All compact $3$-manifolds admit canonical decompos…

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 $G$ 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…

Let $G$ be a finitely generated group of covering translations acting on a Whitehead manifold $W$, with quotient a 3-manifold $M$. A partial-irreducible Haken manifold is a Haken m…

Let $X$ be a closed, connected, irreducible, orientable 3-manifold with infinite fundamental group. Its universal covering space is denoted by $\widetilde{X}$. Universal Covering C…

Let ${\cal H}$ be the Heegaard category, and let $T(V)$ denote the relevant target associated with a vector space $V$. Residual representability conjecture. If $a$ and $b$ are dist…

Let $M_i$ ($i=1,2$) be irreducible 3-manifolds with connected boundary, with $\partial M_1\cong\partial M_2\cong F$. Let $\mathcal{D}_i$ be the set of essential curves in $F$ that…