Cellular homology of complex projective space ($H_{2j}({\mathbb{CP}}^n;\mathbb{Z})=\mathbb{Z},0\le j\le n$)

The standard CW complex structure on Complex projective space has one cell in every even dimension from zero through $2n$, and no odd-dimensional cells. Its cellular chain complex therefore has zero differential. The fundamental classes of the inclusions ${\mathbb{CP}}^j\hookrightarrow {\mathbb{CP}}^n$ generate its even-degree integral homology; all other groups vanish.