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
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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

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