Complex projective plane

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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 ${\pi }_k(S^5)$