Cohomology of complex projective space

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

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

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

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:

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

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

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