Smooth structures on $\mathbb{C}{P}^m$ for $5\le m\le 8$

Abstract

We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex projective m-space $\mathbb{C}{P}^m$ for $m=5,6,7$ and $8$. As an application, for $m=7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}{P}^m$ and obtain a bound on the number of smooth homotopy complex projective m-spaces with given Pontryagin classes up to orientation-preserving diffeomorphism. We also show that there exists a smooth manifold which is tangentially homotopy equivalent but not homeomorphic to $\mathbb{C}{P}^8$.

This manuscript has been substantially revised, resulting in a new version with largely non-overlapping text, although it addresses the same research problem. The revised paper has been submitted separately and is available under the arXiv ID: 2604.27521. In light of this, I kindly ask for your approval to withdraw the earlier version