Almost Complex Structures on Homotopy Complex Projective Spaces

TL;DR

This paper demonstrates that all homotopy complex projective spaces of certain dimensions admit almost complex structures and classifies these structures using Chern classes, providing new proofs and insights.

Contribution

It establishes the existence of almost complex structures on homotopy spaces for 3  n  6 and classifies them by their Chern classes, offering a new proof for spaces of dimension 4.

Findings

All homotopy spaces for 3  n  6 admit almost complex structures.

Classification of these structures by their Chern classes.

A new proof of Libgober and Wood's classification result for spaces of dimension 4.

Abstract

We show that all homotopy $\mathbb{C}P^n$s, smooth closed manifolds with the oriented homotopy type of $\mathbb{C}P^n$, admit almost complex structures for $3 \leq n \leq 6$, and classify these structures by their Chern classes. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy $\mathbb{C}P^4$s.