Cohomology of real 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 real projective space
Get more specific information about real projective space | Get more computations of homology group

Statement

Odd-dimensional projective space with coefficients in integers

$H^p({\mathbb{P}}^n(\mathbb{R}))=\{\begin{array}{cc}\mathbb{Z},& p=0,n\\ \mathbb{Z}/2\mathbb{Z},& p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},0<p<n\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

Even-dimensional projective space with coefficients in integers

$H^p({\mathbb{P}}^n(\mathbb{R}))=\{\begin{array}{cc}\mathbb{Z},& p=0\\ \mathbb{Z}/2\mathbb{Z},& p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},0<p\le n\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$

Cohomology groups with integer coefficients in tabular form

We illustrate how the cohomology groups work for small values of $n$. Note that for $p>n$, all cohomology groups $H^p$ are zero, so we omit those cells for visual ease.

$n$	Real projective space $\mathbb{R}{\mathbb{P}}^n$	Orientable?	$H_0$	$H_1$	$H_2$	$H_3$	$H_4$	$H_5$
1	circle	Yes	$\mathbb{Z}$	$\mathbb{Z}$
2	real projective plane	No	$\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$
3	RP^3	Yes	$\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}$
4	RP^4	No	$\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$
5	RP^5	Yes	$\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}$