Almost Complex Structures on Homotopy Complex Projective Spaces

Keith Mills

Published 2022-01-18, updated 2022-02-07Version 2

We completely answer the question of when a homotopy $\mathbb{C}P^n$, a smooth closed manifold with the oriented homotopy type of $\mathbb{C}P^n$, admits an almost complex structure for $3 \leq n \leq 6$. For $3 \leq n \leq 5$ all homotopy $\mathbb{C}P^n$s admit almost complex structures, while for $n=6$ there exist homotopy $\mathbb{C}P^n$s that do not. 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.