Homology of complex projective space

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology group and the topological space/family is complex projective space
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Statement

The homology of complex projective space is given as follows:

$H_p(\mathbb{C}{P}^n)=\mathbb{Z}p=0,2,4,\dots ,2n$

and zero otherwise.

Related invariants

Betti numbers

The Betti numbers are $1$ for $0,2,4,\dots ,2n$ and $0$ elsewhere.

Poincare polynomial

The Poincare polynomial is given by:

$PX=1+x^2+x^4+\dots +x^{2n}=\frac{x^{2n+2}-1}{x^2-1}$

Euler characteristic

The Euler characteristic is $n+1$.

Proof

Further information: CW structure of complex projective space

We use the CW-complex structure on complex projective space that has exactly one cell in every even dimension till $2n$. The cellular chain complex of this thus has $\mathbb{Z}$s in all the even positions till $2n$, and hence its homology is $\mathbb{Z}$ in all even dimensions till $2n$.