Let $G$ be a finitely presented group,
$$ G = \langle s_1, \ldots, s_n \mid r_1, \ldots, r_m \rangle.$$
Given such a presentation, one constructs the presentation complex $\mathcal{P}$ with a single vertex, by attaching an edge for each generator $s_i$, and a 2-cell for every relator so that the attaching map "reads-off" the relator word $r_j$.
Suppose that $G$ is of type $F_n$. This means that there exists a $K(G,1)$ complex with a finite $n$-skeleton. When is it the case that such a $K(G,1)$ can be obtained from $\mathcal{P}$ by possibly adding cells of higher dimensions? What are the possible obstructions?