3 problems

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

Let $X$ be a compactum, let $n\geq 2$ be an integer, and let $\mathcal G$ be a collection of abelian groups such that $\dim_G X\leq n$ for every $G\in\mathcal G$. Let…

The arbitrary-group acyclic resolution conjecture. There should exist a compactum $Z$ with