Complex projective plane

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

Fill this in later