7 problems

First-type conjecture for Markov compacta. All Markov compacta are of the first type.

Uspenskij's conjecture. Not every $k$-dimensional map $f\colon X\longrightarrow Y$ between compacta admits approximation by $k$-dimensional simplicial maps; equivalently, the answe…

Let $K$ be a countable CW complex. Write $AE(X)$ for the property that every map from a closed subset of $X$ to $K$ extends over $X$. The class of compacta … is said to have a univ…

The p-resolution obstruction conjecture. No compactum $Y$ with

Cardinality conjecture for homogeneous sigma-countably tight compacta. Every infinite homogeneous sigma-countably tight compactum has cardinality $\mathfrak{c}$.

Levin's stable intersection conjecture. Compacta $X$ and $Y$ do not admit a stable intersection in $\mathbb{R}^n$ if and only if

Dranishnikov's product-dimension conjecture. The term $\dim Y+\dim f$ can be replaced by