Homology of complex projective space

This article describes the homology of the following space or class of spaces: complex projective space

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

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 $2n$.