64 problems

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…

Let $M$ be a closed, orientable and atoroidal $3$--manifold. Generalized Waldhausen conjecture. The manifold $M$ has only finitely many Heegaard splittings of any fixed genus, up t…

Let a strongly irreducible Heegaard surface lie in a Riemannian three-manifold. Pitts–Rubinstein conjecture. It is either isotopic to a minimal surface of index at most $1$, or iso…

Gordon's conjecture. The connected sum of two unstabilized Heegaard splittings is unstabilized.

A trisection of a closed 4-manifold is a decomposition into three 4-dimensional 1-handlebodies whose pairwise intersections are 3-dimensional handlebodies and whose triple intersec…

Recall that a Heegaard splitting of a compact $3$-manifold $M$ is a surface $S\subset M$ that divides $M$ into two compression bodies, $M=C_1\cup_S C_2$, where $S$ is the exterior…

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…

Gordon's conjecture. The connected-sum Heegaard splitting $\mathcal{V}\cup_S\mathcal{W}$ is stabilized if and only if at least one of the summand splittings

Let $M_f$ be a hyperbolic $3$-manifold with Heegaard surface $\Sigma$ of minimal genus, and let ${\mathcal D}_1$ and ${\mathcal D}_2$ be the disk sets of the two handlebodies. For…

Johnson's amalgamation conjecture. The group $\operatorname{Isot}(M,\Sigma)$ is an amalgamation of the two groups $\operatorname{Isot}(V^-_{\Sigma'},\Sigma^-)$ and…

Rubinstein's conjecture. $H$ is either isotopic to a minimal surface or isotopic to the oriented double cover of a non-orientable minimal surface with a vertical handle attached.

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 $K_1,K_2\subset S^3$, write $K=K_1\#K_2$, and let $E(K_i)$ denote the exterior of $K_i$. Let $t(K_i)$ denote the tunnel number, and call a Heegaard splitting minimal genus if i…

Let $K_1$ and $K_2$ be knots in $S^3$, and write $K=K_1\#K_2$. Let $E(K)$ denote the exterior of $K$, and let $t(K)$ denote its tunnel number. A Heegaard splitting is weakly reduci…

A knot exterior has a $(g,b)$ presentation if its fundamental group has a presentation with $g+b$ generators, $b$ of which are meridional; a knot has a $(g,b)$ decomposition in the…

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…

Quotient-module conjecture. The module ${\mathcal S}_3(M)$ is always a quotient module of

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 $K\subset S^3$ be a knot that is not $\gamma$--primitive, and let a minimal genus Heegaard splitting of its exterior $E(K)$ be given. A boundary stabilization is the genus-incr…

A Heegaard splitting is a decomposition of a closed orientable $3$-manifold into two handlebodies. The stabilization conjecture. After stabilizing a Heegaard splitting once or poss…

Let $X$ and $Y$ each have a single incompressible boundary component of genus at least two, let $\varphi\colon\partial X\to\partial Y$ be a gluing map, and let $M=X\cup_\varphi Y$.…

Let $X$ have a single boundary component $F$ with $g(F)\geq 2$, and let $V\cup_H W$ be a minimal-genus Heegaard splitting of $X$ with $F$ contained in the compression body $V$. Let…

Let $X$ and $Y$ be 3-manifolds, each with a single incompressible torus boundary component, and let $M$ be obtained by gluing them along their boundaries. There is a complexity on…

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…