137 problems

Local compactness–local solidity conjecture. $X$ is locally compact if and only if

Bryant's modified Bing–Borsuk conjecture. Every locally compact homogeneous ANR-space of dimension $n\geq 3$ is a generalized $n$-manifold. This modifies the Bing–Borsuk conjecture…

Let two spheres in $\mathbb{R}^3$ be equipped with oriented normal bundles and intersect along a circle in the space $\tilde Y$, with the canonical orientation induced on the inter…

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face i…

Let $\mathcal{M}$ be a realizable oriented matroid of rank $r$. Its extension space is the simplicial complex associated to the natural poset of one-point extensions of…

Let $X$ be a closed oriented even $4$-manifold, meaning that its intersection form is even, and let $\sigma(X)$ denote its signature. The $\frac{5}{4}$-conjecture. One has … This i…

Relations-between-relations conjecture. The $2$-complex $C$ has precisely these types of $2$-cells.

For each fixed complex dimension, consider compact hyper-Kähler manifolds and identify two such manifolds when they are homeomorphic, equivalently when they have the same topologic…

Let $\mathcal{F}^{\mathbf R}_{k,n}$ denote the space of real spherical tight frames with $k$ frame vectors in dimension $n$. Connectedness conjecture. For $k,n\in\mathbf{N}$ with…

A continuum is a non-degenerate connected compact metric space. For a metric space, the double midset property (DMP) means that every pair of distinct points has a midset consistin…

Let $S=\mathbb{S}^2$ and let $B$ be the set of crossingless matchings on $2n$ points. For each $a\text{ in }B$, let $S_a$ be the submanifold of $S^{\mathord{\times}2n}$ consis…

Let $F_n$ be the free group on $n>1$ generators. Write $LO(F_n)$ for the space of left-orderings of $F_n$ and $BiO(F_n)$ for the space of bi-orderings, each with the topology inher…

Let $M$ be a complex curve. Let $\gamma_{i}^{\left(1\right)}$, $i\in I^{\left(1\right)}$, be a disjoint family of smooth embedded cycles in $M$ such that…

Let $(M,g)$ be a gravitational instanton. Finite-topological-type conjecture. The manifold $M$ has finite topological type, meaning that it can be compactified into a smooth manifo…

Let $t_1$ and $t_2$ be tame tangles, and let $T_1$ and $T_2$ be arbitrarily chosen mosaic representatives of $t_1$ and $t_2$, respectively. Topological equivalence conjecture for t…

Let $\mathcal N_n(\mathbb R^2)$ be the moduli space of planar graphs with $n$ bounded faces, and let $\operatorname{uConf}_n(\mathbb R^2)$ be the configuration space of $n$ unlabel…

Let $f\taut$ be a map, where $\taut$ denotes the boundary of the space of embeddings, and let $\tP(f)$ denote the corresponding pseudo-isotopy space. Local weak contractibility con…

Domokos–Horváth–Goriely–Regős conjecture. Every polyhedral tiling can be completely softened.

Index-bound conjecture. One has

Minimal vertex-count conjecture. (i) For a CCP of a topological torus without self-intersection, the minimal number of vertices is $9$. (ii) For an orientable CCP of genus $g$, if…

Let $d=d'd”$, where $d'$ is the minimal prime factor of $d$ and $d”>2$, so that $d$ is neither prime nor equal to $4$. Let $\mathcal{D}$ be branch data with $k$ partitions, and sup…

Let $d$ be a positive integer, and let $\mathcal{D}$ be a collection of nontrivial partitions of $d$. A collection satisfying the Riemann–Hurwitz formula is called a candidate bran…

Hopf-Chern conjecture.

Let $M$ be an $n$-dimensional projective manifold and let $D$ be a smooth ample divisor on $M$. Suppose that the natural homomorphism … is bijective for $0\leq k\leq 2(n-1)$. Fujit…

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathbb{X}'$ be an independent copy of $\mathbb{X}$. A neighborhood of $\mathbb{X}$ means a neighborhoo…