# Cohomology of fine Grassmannian manifold

Let it consider the fibration $U(n) \rightarrow V_{n}(\mathbb{C}^{k}) \rightarrow Gr(n,k)$ where $U(n)$ is the unitary group, $V_{n}(\mathbb{C}^{k})$ the Stiefel manifold and $Gr(n,k)$ the Grassmannian manifold. Now I want to use a spectral sequence in order to calculate the coohomology of finite Grassmannian manifold. So how can I calculate $H^{s}(V_{n}(\mathbb{C}^{k}))$ and $H^{m}(U(n))$?

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To calculate the cohomology or homology of spaces in a fibration you can use the Serre spectral sequence. For $U(n)$ one can use the fibration $U(n−1)\rightarrow U(n)\rightarrow S^{2n-1}$ which is a special case of the fibration $V_{n}(\mathbb{C}^{k})\rightarrow V_{n+1}(\mathbb{C}^{k+1})→S^{2k−1}$ when $k=n$. –  Callan McGill Feb 2 at 23:54
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