= Cellular free resolution from a contractible universal cover
For a connected <CW complex> $X$ with contractible <universal cover>, let $\Gamma=\pi_1(X)$ and $R=\mathbb Z\Gamma$. Lift each cell once and transport its orientation by <deck transformations>. Its translates form an $R$-basis of the cellular chains of the cover. The augmentation sends every vertex to one, and contractibility makes the augmented chain complex an exact <free resolution> of the trivial module $\mathbb Z$. An $R$-linear cochain with trivial coefficient action is constant on each orbit of lifts, identifying $\operatorname{Hom}_R(C_*(\widetilde X),\mathbb Z)$ with the <cellular cohomology> complex of $X$. Consequently $H^i(X;\mathbb Z)=\operatorname{Ext}_R^i(\mathbb Z,\mathbb Z)$. The same argument with singular simplex lifts works for spaces with a contractible universal cover.
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