Geodesic
Cohomology of unitary and Sympletic groups
Lobachevskii journal of mathematics, Tome 5 (1999), pp. 57-60

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We compute the cohomology rings of $U(n)$ and $Sp(n)$ and of their Stiefel varieties by using the Serre spectral sequence. This approach is much simpler than the usual method, that of using the cell structures. The argument here also shows that the cohomology of $U(n)$ is built from those of $U(n-1)$ and $S^{n-1}$ through a fiber bundle; a similar result holds for $Sp(n)$. 
@article{LJM_1999_5_a3,
     author = {D. Yau},
     title = {Cohomology of unitary and {Sympletic} groups},
     journal = {Lobachevskii journal of mathematics},
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     publisher = {mathdoc},
     volume = {5},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_1999_5_a3/}
}
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D. Yau. Cohomology of unitary and Sympletic groups. Lobachevskii journal of mathematics, Tome 5 (1999), pp. 57-60. http://geodesic.mathdoc.fr/item/LJM_1999_5_a3/