Homology of complex projective space

Statement

The homology of complex projective space is given as follows:

$H_{p}(\mathbb {C} P^{n})=\mathbb {Z} \qquad p=0,2,4,\ldots ,2n$

and zero otherwise.

Related invariants

Betti numbers

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

Poincare polynomial

The Poincare polynomial is given by:

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

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 $2n$.