4 problems

The homological $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname{FP}_n$, $G_1$ and $G_2$ are of type $\operatorname{FP}_{n+1}$, and $Q$ is of type…

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname F_n$, $G_1$ and $G_2$ are of type $\operatorname F_{n+1}$, and $Q$ is of type $\operatorname F_{n+2}$, then…

Let $N_1 \to G_1 \to Q$ and $N_2 \to G_2 \to Q$ be short exact sequences. Assume that $Q$ is of type $F_{n+2}$, $G_1$ and $G_2$ are groups of type $F_{n+1}$, and $N_1$ is of type…

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_1$ is of type $\operatorname{F}_n$, both $\Gamma_1$ and $\Gamma_2$ are of type $\operatorname{F}_{n+1}$, and $Q$ is of type…