Homology of real projective space

Statement

Even-dimensional projective space

$H_p(\mathbb{R}{P}^n)=\mathbb{Z}p=0,n$

For odd $p$ with $0<p<n$:

$H_p(\mathbb{R}{P}^n)=\mathbb{Z}/2\mathbb{Z}$

And zero otherwise.

Odd-dimensional projective space

We have:

$H_p(\mathbb{R}{P}^n)=\mathbb{Z}p=0$

For odd $p$ with $0<p<n$:

$H_p(\mathbb{R}{P}^n)=\mathbb{Z}/2\mathbb{Z}$

And zero otherwise.

Thus the key difference between even and odd dimensional projective spaces is that the top homology vanishes in even-dimensional projective spaces. This is related to the fact that even-dimensional projective space is non-orientable, while odd-dimensional projective space is orientable.

Related invariants

Betti numbers

The Betti numbers of real projective space are thus $1$ at $0$ and at $n$ if $n$ is odd, and $1$ only at $0$ if $n$ is even.

Poincare polynomial

The Poincare polynomial of real projective space is $1+x^n$ if $n$ is odd, and $1$ if $n$ is even.

Euler characteristic

The Euler characteristic is 0 if $n$ is odd and $1$ if $n$ is even.

Relation with the sphere

There is a double cover from the $n$-sphere to real projective $n$-space. This double cover induces an isomorphism on all even-dimensional homologies (and of course on all homologies higher than $n$). Fill this in later