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

Statement

With coefficients in the integers

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

With coefficients in an abelian group or module

For coefficients in an abelian group $M$, the homology groups are:

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

Cohomology ring structure

With coefficients in the integers

The cohomology ring with coefficients in the integers is given as:

$H^{*}({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})=\mathbb{Z}[x]/(x^{n+1})$

where the following are true:

• The base ring of coefficients is identified with $H^0$.
• $x$ (or rather, its image mod $x^{n+1}$) is identified additive generator for $H^2({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})$. Note that we could pick either of the two additive generators, since the group is isomorphic to $\mathbb{Z}$.
• Each $x^j$ is identified with an additive generator for $H^{2j}({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})$. In particular, $x^n$ is identified with a generator for the top cohomology, or a fundamental class in cohomology.

Here are some additional observations:

• The only ring automorphisms of $H^{*}({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})$ arising from self-homeomorphisms of the complex projective space are the identity map and the automorphism that acts as the negation map on $H^2$ and induces corresponding multiplication by $(-1{)}^j$ maps on each $H^{2j},0\le j\le n$.
• In particular, this means that if $n$ is even, then the top cohomology class is rigid under automorphisms, i.e., there is no automorphism that acts as the negation map on the top cohomology.
• We get a map:

$\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}({\mathbb{P}}^n(\mathbb{C}))\to \{\pm 1\}\cong \mathbb{Z}/2\mathbb{Z}$

which sends a homeomorphism to 1 if it acts as the identity on $H^2$ and -1 otherwise.

• More generally, for any continuous self-map of ${\mathbb{P}}^n(\mathbb{C})$, we can find an integer $m$ such that this map induces multiplication by $m$ on $H^2$. Consequently, it induces multiplication by $m^j$ maps on each $H^{2j},0\le j\le n$.

Facts proved using the cohomology ring structure

• Complex projective space has fixed-point property iff it has even complex dimension
• Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension