nLab cohomology of classifying spaces

Ordinary cohomology

Computed by Borel in 1950s: 1, 2.

Differential cohomology

References

• Armand Borel, Sur La Cohomologie des Espaces Fibres Principaux et des Espaces Homogenes de Groupes de Lie Compacts, Annals of Mathematics 57:1 (1953), 115.

• Armand Borel, Sur L’Homologie et la Cohomologie des Groupes de Lie Compacts Connexes, American Journal of Mathematics 76:2 (1954), 273.

Integral coefficients:

• Edgar H. Brown, The cohomology of $BSO_n$ and $BO_n$ with integer coefficients, Proceedings of the American Mathematical Society 85:2, 283-288. doi.

• Mark Feshbach?, The Integral Cohomology Rings of the Classifying Spaces of O(n) and SO(n), Indiana Univ. Math. J. 32 No. 4 (1983), 511–516. doi.

• Feifei Fan, The cohomology of the classifying space of $PU(4)$, arXiv.

Differential cohomology:

• Ivan Kolář, Gauge-Natural Forms of Chern-Weil Type, Annals of Global Analysis and Geometry 11: 41–47, 1993.