15 problems

Let $f:X\to B$ be a cellular map of finite CW complexes. Let $P$ be the finite-space map from the three-point diagram with one distinguished point and two indistinguishable points…

Let $\partial\Delta^n$ be a finite topological space weakly homotopy equivalent to the sphere $\mathbb S^n$, with $\mathbb S^n\to\partial\Delta^n$ a trivial Serre fibration, and le…

Let $K$ be a topological space equipped with maps $K\to\{\bullet\}$ lying in both indicated right-orthogonal classes: the class generated by the finite-space maps…

Let $f$ be a map of finite topological spaces. Say that a fibre has two points bounded above but not below when its two points have a common upper bound but no common lower bound,…

Let $\{o\}\to\{o\to c\}$ be the indicated map of finite topological spaces, and let the subscript $<5$ denote the subclass consisting of maps between spaces with fewer than $5$ poi…

Let $f:X\to Y$ be a map of finite topological spaces, and let $|f|_\omega:|X|_\omega\to|Y|_\omega$ be its non-$T_0$ geometric realisation. Let $\mathsf{W}\to\mathsf{V}$ and…

Let $Y$ be a finite topological space, and let $|Y|_\omega$ denote its non-$T_0$ geometric realisation. Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the finite-s…

Let $X$ be a finite topological space, let $PX$ be its path space, and let $\operatorname{tc}(X)$ denote its topological complexity. Let $\operatorname{cat}(X)$ denote its Lusterni…

Let $\mathbb{S}^1_{2^k}$ be the Khalimsky circle with $2^k$ points, where $k\geq 1$, and let $\operatorname{tc}(X)$ denote the topological complexity of a space $X$. Tanaka's conje…

Stronger Webb conjecture. The finite space

Uniqueness conjecture. If the reconstruction problem has more than one solution up to homeomorphism, then trivial clans rooted at $k$-sets for some $k<n-1$ must exist.

May's conjecture. For every $n\in\mathbb{N}$, $\mathbb{S}^n S^0$ is a minimal finite model of the $n$-sphere.

Torsion lower-bound conjecture. If $H_k(X)$ has torsion, then

Hardie–Vermeulen–Witbooi conjecture. The integral homology of $X$ has no $2$-torsion.

May's conjecture. The finite model $\mathbb{S}^n S^0$ is minimal.