Complex projective plane

From Topospaces
Jump to: navigation, search
This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

The complex projective plane is the complex projective space of complex dimension 2. As a manifold over the reals, it has dimension 4. It is denoted \mathbb{C}\mathbb{P}^2 or \mathbb{P}^2(\mathbb{C}).

Alternatively, it can be viewed as the quotient of the space S^2(\mathbb{C}) \cong S^5 under the action of S^0(\mathbb{C}) \cong S^1 by multiplication. In particular, there is a fibration S^1 \to S^5 \to \mathbb{C}\mathbb{P}^2.

Algebraic topology

Homology groups

Further information: homology of complex projective space

The homology groups with coefficients in \mathbb{Z} are as follows: H_0(\mathbb{C}\mathbb{P}^2) \cong H_2(\mathbb{C}\mathbb{P}^2) \cong H_4(\mathbb{C}\mathbb{P}^2) \cong \mathbb{Z}, and all other homology groups are zero.

More generally, the homology group with coefficients in a module M over a commutative unital ring R are as follows: H^0(\mathbb{C}\mathbb{P}^2;M) \cong H^2(\mathbb{C}\mathbb{P}^2;M) \cong H^4(\mathbb{C}\mathbb{P}^2;M) \cong M, and all other homology groups are zero.

Cohomology groups

Further information: cohomology of complex projective space

The cohomology groups with coefficients in \mathbb{Z} are as follows: H^0(\mathbb{C}\mathbb{P}^2) \cong H^2(\mathbb{C}\mathbb{P}^2) \cong H^4(\mathbb{C}\mathbb{P}^2) \cong \mathbb{Z}, and all other cohomology groups are zero. The cohomology ring is \mathbb{Z}[x]/(x^3) where x is an additive generator for the second cohomology group.

More generally, the cohomology group with coefficients in a commutative unital ring R are as follows: H^0(\mathbb{C}\mathbb{P}^2;R) \cong H^2(\mathbb{C}\mathbb{P}^2;R) \cong H^4(\mathbb{C}\mathbb{P}^2;R) \cong R, and all other cohomology groups are zero. The cohomology ring is R[x]/(x^3) where x is a R-module generator for the second cohomology module.

Homotopy groups

Further information: homotopy of complex projective space

The homotopy groups are as follows:

Value of k General name for homotopy group/set \pi_k What is \pi_k(\mathbb{C}\mathbb{P}^n for generic n \ge 2?) What is \pi_k(\mathbb{C}\mathbb{P}^2)?
0 set of path components one-point set one-point set, so \mathbb{C}\mathbb{P}^2 is a path-connected space
1 fundamental group trivial group trivial group, so \mathbb{C}\mathbb{P}^2 is a simply connected space.
2 second homotopy group \mathbb{Z} \mathbb{Z}
3 third homotopy group trivial group trivial group
4 fourth homotopy group trivial group trivial group
5 fifth homotopy group \mathbb{Z} if n = 2, zero otherwise \mathbb{Z}
k \ge 6 Same as \pi_k(S^5)