6 problems

Van Kampen obstruction conjecture. The van Kampen obstruction vanishes if and only if $\varphi$ is approximable by embeddings.

Failure conjecture. The analogues of these criteria for $\varphi$ are false.

Let $X$ be a compact, locally connected, metrizable space that is locally embeddable in $S^2$. A simple set of circles in $X$ is a set $S$ of circles with the simplicity property u…

Graph-map criterion conjecture. The map $\varphi$ is approximable by embeddings if and only if

Disjoint-arc criterion. The path $\varphi$ is approximable by embeddings if and only if, for every pair of disjoint arcs $I_1,I_2\subset I$, the restrictions…

Generalized graph-map conjecture. Theorem1.5 is true for $\varphi$.