Let $G$ be group. Each cell partition of the universal cover of $K(G, 1)$ delivers a (projective?) resolution of $\mathbb{Z}$ over group algebra $\mathbb{Z}G$.
Can one construct a pair of contractable cell complex and it's surjective map onto $K(G, 1)$, homotopically equivalent to universal cover, corresponding to given resolution of $\mathbb{Z}$ over $\mathbb{Z}G$?
Or possibly at least a contractable cell complex with action of $G$ on it?