We compute the cohomology of the unordered configuration spaces of the sphere $S^2$ with integral and with $\mathbb{Z}/p \mathbb{Z}$-coefficients using a cell complex by Fuks, Vainshtein and Napolitano.
Homological stability for unordered configuration spaces
Homology and cohomology of cubical sets with coefficients in systems of objects
The cohomology ring of unordered configuration spaces of the torus
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