# Cohomology of complex projective space

From Topospaces

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

## Contents

## Statement

### With coefficients in the integers

### With coefficients in an abelian group or module

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

## Cohomology ring structure

### With coefficients in the integers

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

where the following are true:

- The base ring of coefficients is identified with .
- (or rather, its image mod ) is identified additive generator for . Note that we could pick either of the two additive generators, since the group is isomorphic to .
- Each is identified with an additive generator for . In particular, 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 arising from self-homeomorphisms of the complex projective space are the identity map and the automorphism that acts as the negation map on and induces corresponding multiplication by maps on each .
- In particular, this means that if 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:

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

- More generally, for any continuous self-map of , we can find an integer such that this map induces multiplication by on . Consequently, it induces multiplication by maps on each .