72 problems

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…

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. The algebraic girth is defined using the minimum girth of a nontrivial relato…

A knot manifold is an irreducible, orientable, compact 3-manifold whose boundary is a single torus. It is hyperbolic if its interior admits a complete hyperbolic metric of finite v…

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…

A knot exterior $X$ is the complement of an open tubular neighborhood of a knot, and $g(X)$ denotes its Heegaard genus. An exterior admits a primitive meridian if it has a minimal-…

Let $M$ be a closed, orientable three-manifold. A Heegaard splitting is full when it does not have the disjoint curve property in the terminology of the source. Finiteness conjectu…

Let $M$ be a closed, orientable, atoroidal three-manifold. A Heegaard splitting is strongly irreducible if every compressing disk on one side intersects every compressing disk on t…

Gilmer's cut-number conjecture. For every closed, connected $3$-manifold $M$,

Let $M$ be a closed, oriented $3$-manifold, and let $\mathscr S(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm1}]$. A module decomposition into free modules an…

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 cusped hyperbolic manifold with at least one cusp, and let $\mathbf{s}$ and $\mathbf{s'}$ be sets of slopes on the cusps. Futer–Purcell–Schleimer's conjecture. If the…

Let $K$ be a knot in an L-space. The knot $K$ is persistently foliar if, except for one meridional slope, every boundary slope of its complement is strongly realized by a co-orient…