Diffeomorphism Classification of Smooth Structures and Tangential Homotopy Types of $\mathbb{C}P^m$ for $5 \le m \le 8$

TL;DR

This paper classifies smooth manifolds homeomorphic to complex projective spaces for dimensions 5 to 8, using advanced techniques to distinguish their smooth structures and tangential homotopy types.

Contribution

It provides a detailed diffeomorphism classification of manifolds related to $ ext{CP}^m$ for 5 ≤ m ≤ 8, including new results on their smooth structures and homotopy types.

Findings

Unique smooth manifold for $ ext{CP}^4$ up to diffeomorphism with the same tangential homotopy type.

Exactly two non-diffeomorphic manifolds for $ ext{CP}^8$ with the same tangential homotopy type.

Classification achieved via concordance classes, surgery sequences, and stable homotopy theory techniques.

Abstract

This paper provides a diffeomorphism classification of smooth manifolds homeomorphic to the complex projective space $\mathbb{C}P^m$ for $m \in \{5, 6, 7, 8\}$. The classification is obtained by computing the group of concordance classes of smooth structures on $\mathbb{C}P^m$ and determining the orbit space under the action induced by the group of self-homeomorphisms. Using these computations in conjunction with the tangential surgery exact sequence and techniques from stable homotopy theory, we determine the diffeomorphism classes of smooth manifolds within the tangential homotopy type of $\mathbb{C}P^m$ for $4 \le m \le 8$. We also investigate the relationship between these two classification problems by studying the natural map from the homeomorphism type to the tangential homotopy type. As a consequence, we prove that for $m = 4$, there exists a unique smooth manifold, up to diffeomorphism, that is tangentially homotopy equivalent to $\mathbb{C}P^4$ but not homeomorphic to it. Furthermore, for $m = 8$, there exist exactly two pairwise non-diffeomorphic smooth manifolds that are tangentially homotopy equivalent to $\mathbb{C}P^8$ but not homeomorphic to it.