76 problems

Let $M$ be a compact orientable simple $3$-manifold, let $F$ be a component of $\partial M$ with genus at least $2$, and let $\alpha,\beta$ be separating slopes on $F$. In the redu…

For every integer $g\ge 2$, there are only finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds having a fiber of genus $g$.

For every finite-volume cusped hyperbolic $3$-manifold $M$, there exists a finite ideal triangulation of $M$ whose tetrahedra are realized as positively oriented geodesic ideal tet…

For every closed orientable hyperbolic $3$-manifold $M$, the odd-level $SO(3)$ Turaev–Viro invariants satisfy …

Let $K$ be a hyperbolic knot in $S^3$, let $E_K=S^3\setminus\nu(K)$ be its exterior, and let $x(K)$ denote its Thurston norm, equivalently … Let $\rho^{\mathrm{geom}}$ be a lift of…

Thurston's virtual fibration conjecture. Every finite-volume hyperbolic $3$-manifold has a finite cover that is fibered.

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 $M$ be a hyperbolic knot manifold, meaning a compact, connected, orientable $3$-manifold with torus boundary whose interior admits a complete finite-volume hyperbolic structure…

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 $M_1$ 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.…

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 $M$ be a closed aspherical $3$-manifold and let $\Gamma=\pi_1(M)$. The paper's statements numbered through concern properties of $M$ and $\Gamma$ introduced earlier in the pape…

A cosmetic surgery is a pair of Dehn fillings on a knot or 3-manifold exterior that produce homeomorphic filled manifolds; it is truly cosmetic when the homeomorphism preserves the…

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 $M^{3}$ 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…

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 $X$ 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…

Geometric presentation conjecture. The group $\langle x_1,\ldots,x_n\mid w\rangle$ is the fundamental group of an orientable 3-manifold if and only if $w$ is a geometric word.

Let $(M,W)$ be an admissible pair, where $W$ is a genus two handlebody with meridian disks $\alpha$ and $\beta$, and suppose that $M-W$ is $\partial$-irreducible. Boundary-irreduci…

Let $(M,W)$ be an admissible pair, with $W$ a genus two handlebody and $\alpha,\beta$ the specified meridian disks of $W$; the source defines admissibility and alignment in the pre…

Let $\Sigma$ be a closed, orientable $3$-manifold. Write $\lambda_{\operatorname{PSL}_2(\mathbb C)}(\Sigma)$ and $\lambda_{\operatorname{SL}_2(\mathbb C)}(\Sigma)$ for the respecti…

Let $H$ be a handlebody and let $S$ be a compact surface in $\partial H$. Let $\mathcal{N}(\partial S)$ be the subgroup of $\pi_1(S)$ normally generated by the boundary curves of…

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. A closed curve in $\partial H$ is primitive in $H$ if it does not represent a…