Let $E$ be the tangent bundle of $\mathbb{C}\mathbb{P}^n$. I want to compute the homology of $S(E)$ (the unit tangent bundle).
By the Gysin sequence, I am able to compute the cohomology groups. However, I cannot apply Poincaré duality, because S(E) is a manifold with boundary (I have not learned the theorem for manifolds with boundary).
Can anyone give me a clue, please?