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

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