Is it an easy problem computing Betti numbers of compact homogenous spaces?
I want to compute Betti numbers of well-known compact Riemannian homogenous spaces like $\mathsf{Sp}(2)/\mathsf{SU}(2)$ and $\Big(\mathsf{SU}(3)\times \mathsf{SO}(3)\Big)/\mathsf{U}^\star(2)$ where $\mathsf{U}^\star(2)$ is the image under the embedding $(\iota; \pi): \mathsf{U}(2)\hookrightarrow \mathsf{SU}(3) \times \mathsf{SO}(3)$ given by the natural inclusion and the projection $\pi:\mathsf{U}(2)\rightarrow\mathsf{U}(2)/\Bbb S^1\cong \mathsf{SO}(3)$.
I have seen constructing exact sequence using fiber bundles but I don't remember the exact procedure.
How to compute Betti numbers of such compact homogenous spaces?
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