Homology of real projective space

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

Statement

Odd-dimensional projective space with coefficients in integers

$H_{p} (P^{n} (R)) = Z p = 0, n$

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

$H_{p} (P^{n} (R)) = Z / 2 Z$

And zero otherwise.

Even-dimensional projective space with coefficients in integers

We have:

$H_{p} (P^{n} (R)) = Z p = 0$

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

$H_{p} (P^{n} (R)) = Z / 2 Z$

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.

Coefficients in a 2-divisible ring

If we consider the homology with coefficients in a module $M$ over a ring $R$ where 2 is invertible, the we have: $H_{n} (P^{n} (R); M) ≅ H_{0} (P^{n} (R); M) ≅ M$ (regardless of whether $n$ is even or odd) and all other homology groups are zero.

In particular, these results are valid over the field of rational numbers or over any field of characteristic zero.

Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant	General description	Description of value for spheres
Betti numbers	The $n^{t h}$ Betti number $b_{k}$ is the rank of the $k^{t h}$ homology group.	$b_{0} = 1$ . $b_{n} = 1$ if $n$ is odd and $b_{n} = 0$ if $n$ is even.
Poincare polynomial	Generating polynomial for Betti numbers	$1 + x^{n}$ if $n$ is odd. $1$ if $n$ is even.
Euler characteristic	$\sum_{k = 0}^{\infty} (- 1)^{k} b_{k}$	$0$ if $n$ is odd. $1$ if $n$ is even. Note that the Euler characteristic is half the Euler characteristic of $S^{n}$ , which is its double cover.

Facts used

CW structure of real projective space

Proof

The proof follows from fact (1). See more details on that page.