The classification of smooth structures on a homotopy complex projective space

Abstract

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective $n$-space $\mathbb{C}{\text{P}}^n$, where $n=3$ and $4$. Let $M^{2n}$ be a closed smooth $2n$-manifold homotopy equivalent to $\mathbb{C}{\text{P}}^n$. We show that, up to diffeomorphism, $M^6$ has a unique differentiable structure and $M^8$ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover $N^{2n}$ of $\mathbb{C}{\text{P}}^n$ for $n=4,7$ or $8$ and six distinct differentiable structures on $N^{10}$.