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

$H^{p}(\mathbb {P} ^{n}(\mathbb {C} );\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad p\ \operatorname {even} ,0\leq p\leq 2n\\0,&\qquad \operatorname {otherwise} \\\end{array}}\right.$

With coefficients in an abelian group or module

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

$H^{p}(\mathbb {P} ^{n}(\mathbb {C} );M)=\left\lbrace {\begin{array}{rl}M,&\qquad p\ \operatorname {even} ,0\leq p\leq 2n\\0,&\qquad \operatorname {otherwise} \\\end{array}}\right.$

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 $H^{2j},0\leq j\leq n$.
• 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:

$\operatorname {Homeo} (\mathbb {P} ^{n}(\mathbb {C} ))\to \{\pm 1\}\cong \mathbb {Z} /2\mathbb {Z}$

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 $H^{2j},0\leq j\leq n$.

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