2 problems

Let $M$ be a $2n$-dimensional compact oriented manifold homotopy equivalent to the complex projective space $4\mathbb{CP}^n4$. Suppose that $M$ admits a non-trivial $S^1$-action. P…

Let $k$ be a positive integer, and let $\operatorname{Conf}^n\mathbb{C}P^k$ denote the configuration space of $n$ ordered distinct points in complex projective $k$-space. Write…