21.146 (2026)

(Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial. A group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example, type $F_1$ is equivalent to finite generation, and type $F_2$ is equivalent to finite presentability. Type $F_\infty$ means type $F_n$ for all $n$.

For $n \geqslant 3$, does every group of type $F_{n-1}$ embed as a subgroup of a group of type $F_n$? Or even in a group of type $F_\infty$?