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

## TL;DR

This paper classifies smooth manifolds homeomorphic to complex projective spaces for dimensions 5 through 8, analyzing their diffeomorphism types and tangential structures.

## Contribution

It provides a classification of smooth structures on $ ext{CP}^m$ for $m=5,6,7,8$, including computations of tangential structure sets and existence of exotic manifolds.

## Key findings

- Classified all smooth manifolds homeomorphic to $ ext{CP}^m$ for $m=5,6,7,8$.
- Computed the smooth tangential structure set for $ ext{CP}^7$ and $ ext{CP}^8$.
- Demonstrated existence of a tangentially homotopy equivalent but non-homeomorphic manifold to $ ext{CP}^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 $\mathbb{C}P^{8}$.