Almost complex structures on homotopy complex projective spaces

Abstract

We show that all homotopy CPns, smooth closed manifolds with the oriented homotopy type of CPn, admit almost complex structures for 3≤n≤6, and classify these structures by their Chern classes for n=4,6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP4s.