The classification of smooth structures on a homotopy complex projective space

TL;DR

This paper classifies smooth structures on manifolds homotopy equivalent to complex projective spaces, showing uniqueness in some cases and multiple structures in others, advancing understanding of differentiable structures on these manifolds.

Contribution

It provides a classification of smooth structures on certain homotopy complex projective spaces, identifying cases with unique or multiple differentiable structures.

Findings

M^6 has a unique differentiable structure

M^8 has at most two differentiable structures

Multiple structures exist on finite covers of complex projective spaces

Abstract

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective $n$-space $\mathbb{C}\textbf{P}^n$, where $n=3$ and $4$. Let $M^{2n}$ be a closed smooth $2n$-manifold homotopy equivalent to $\mathbb{C}\textbf{P}^n$. 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 $\mathbb{C}\textbf{P}^n$ for $n=4, 7$ or $8$ and six distinct differentiable structures on $N^{10}$.