Complex projective plane

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 $$S^1 \to S^5 \to \mathbb{C}\mathbb{P}^2$$.

Homology groups
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
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
The homotopy groups are as follows: