Cohomology of complex projective space

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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} \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.

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:

\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