= Cellular homology of complex projective space
{title2=$H_{2j}(\mathbb{CP}^n;\mathbb Z)=\mathbb Z,\quad 0\le j\le n$}
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.
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