10 problems

Let $K$ and $L$ be simplicial complexes, and let $\vartheta$ denote their van Kampen obstruction. Algebraic Menger conjecture. If … then … This is presented as a second algebraic n…

Let $K$ and $L$ be simplicial complexes, and let $\vartheta$ denote their van Kampen obstruction. Algebraic Grünbaum–van Kampen–Flores conjecture. If … then … The conjecture is an…

For a countable ordinal $\alpha<\omega_1$, consider the class of functions that are not of Baire class $\alpha$, ordered by topological embeddability. A finite basis for an upward-…

Let $[\omega]^2_+$ denote the relevant domain of ordered pairs of natural numbers, let $C([\omega]^2_+,\mathbb{Q})$ be the space of continuous functions from this domain to…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,2,\ldots,2)$-admissible order, meaning that the least two vertices of each color class form an initial segment o…

Let $K$ be a $d$-dimensional balanced simplicial complex, meaning that its vertices are colored with $d+1$ colors so that every edge has vertices of different colors. Let $K^b$ den…

Let $K$ be a $2$-dimensional simplicial complex, and let $K^s$ denote its symmetric algebraic shifting. Symmetric-shifting conjecture. If $K$ is embeddable in $\mathbb{R}^4$, then…

Embeddability-complexity conjecture. The set $K$ is $\Sigma^0_3$-complete. Equivalently, deciding whether $\overline{P}_i$ embeds into $\overline{P}_j$ for finite presentations…

Symmetric shifting obstruction conjecture. If

Missing-face embeddability conjecture. The complex $\mathsf{K}$ does not embed in the $2d$-sphere.