Almost Complex Structures on Homotopy Complex Projective Spaces

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.