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 $n\geq 2$?)	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	$\mathbb {Z}$
$k\geq 6$			Same as $\pi _{k}(S^{5})$