28 problems

Boundary-determination conjecture. If $\partial X$ and $\partial X'$ are diffeomorphic, then $X$ and $X'$ are diffeomorphic.

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…

Handlebody basis conjecture. For every handlebody $H_n$, ${\mathcal S}_3(H_n)$ is the free $Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}]$-module with basis

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 $M$ be a $3$-manifold, let $W\subset M$ be a genus two handlebody, and let $\alpha$ and $\beta$ be meridian disks of $W$. The exterior $\overline{M-W}$ is irreducible, and…

Let $H_g$ be a genus-$g$ handlebody, and let $\tau$ be a one-vertex triangulation of its boundary. A $[\tau]$-layered-triangulation is a layered-triangulation of $H_g$ extending…

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 $H$ be a handlebody and let $ewlinegamma$ be a simple closed curve in $partial H$. The girth of $gamma$ is the minimum geometric intersection number between $gamma$ and an…

Let $G$ be a finite group, let $\mu(G)$ denote the minimum number of generators of $G$, and let $n$ be an integer with $n>\mu(G)$. A generating $n$-vector of $G$ is an $n$-tuple of…

Let $H$ be a handlebody with invariant measured lamination $\boldsymbol{\Lambda}$, and let the growth rate refer to the associated irreducible automorphism. A lamination is tight w…

Let $\mathcal{H}$ be a genus $2$ handlebody with three curves $\alpha$, $\beta$, and $\gamma$ under consideration, and let $\rho$ be a finite-covolume representation of its fundame…

A ribbon surface is a ribbon surface carried as a $2$-dimensional sub-$2$-handlebody of a $4$-dimensional handlebody. A 1-isotopy is a $(1,2)$-deformation between ribbon surfaces,…

A connected $4$-dimensional $2$-handlebody is a smooth $4$-manifold obtained from a single copy of $D^4$ by attaching finitely many $1$-handles and $2$-handles. Two such handlebodi…

Let $\Bbbk\mathbf{gr}^{\operatorname{op}}$ be the opposite-category module category associated with the handlebody Jacobi-diagram category, and let $A^{L}_d Q$ denote the submodule…

Let $W$ and $W'$ be $4$-dimensional $2$-handlebodies. Two such handlebodies are $2$-equivalent when they are related by a finite sequence of isotopies, handle slides, and creations…

Let $H_g$ be the handlebody of genus $g$, and let $(H_g,cmathcal C)$ be an attaching system, with $H_g^{cmathcal C}$ obtained by attaching $2$-handles along the curves in…

Handlebody conjecture. Every three-dimensional self-affine tile that is a topological three-manifold is homeomorphic to a handlebody.

Let $H$ be a handlebody with $m\geq 3$ spots. The disk graph of $H$ has vertices given by isotopy classes of essential disks, with edges joining classes admitting disjoint represen…

Let $M^m$ be a connected topological manifold, and let $V_0$ be a relatively compact non-empty open subset of $M$ whose boundary is a manifold. An increasing family of relatively c…

Let $H_n$ $H_m$ denote the connected sum of handlebodies of genera $n$ and $m$, let $\mathcal{S}_{2,\infty}(H_2)$ be the Kauffman bracket skein module of the genus-two handlebody,…

Let $H^i_0$ and $H^i_1$ be the handlebodies referred to in the source's preceding remark, and let $g$ denote the relevant genus. Linear boundary-length conjecture. There exists…

For each $g\geq 0$, let $V_0,V_1\subset S^4$ be 3-dimensional genus-$g$ handlebodies with the same boundary, $\partial V_0=\partial V_1$. A simple closed curve in $\partial V_0$ is…

Closure under disk surgery conjecture. For higher genus Heegaard splittings of the $3$-sphere, the primitive disk complex $\mathcal P(V)$ is closed under the disk surgery operation…

Let $\bfF$ denote the category of finitely generated free groups. Consider the functor … that assigns to each bottom tangle $T:m\to n$ the homomorphism $(i_T)_*:F_n\to F_m$. The fu…