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 cohomology group and the topological space/family is real projective space
Get more specific information about real projective space | Get more computations of cohomology group

Statement

Odd-dimensional projective space with coefficients in integers

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

Even-dimensional projective space with coefficients in integers

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

Odd-dimensional projective space with coefficients in an abelian group

For an abelian group $M$, the cohomology is given by:

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

Here, $T$ denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

Even-dimensional projective space with coefficients in an abelian group

For an abelian group $M$, the cohomology is given by:

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

Here, $T$ denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

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	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}$

Facts used

1. Homology of real projective space
2. Dual universal coefficients theorem
3. CW structure of real projective space

Proof using homology groups

Case of odd dimension

Fill this in later

Case of even dimension

Fill this in later

Proof using cochain complex constructed from CW structure

Fill this in later