Complex projective space

From Topospaces
Jump to: navigation, search

Definition

Short definition

Complex projective space is defined as projective space over a field of complex numbers \mathbb{C}, where the topological structure is induced from the structure of the field of complex numbers as a topological field.

Finite-dimensional

Complex projective space of dimension n, denoted \mathbb{C}\mathbb{P}^n or \mathbb{P}^n(\mathbb{C}), is defined as the quotient space under the group action \mathbb{C}^{n+1} \setminus \{ 0 \}/\mathbb{C}^* where \mathbb{C}^* acts by scalar multiplication. It is equipped with the quotient topology.

As a set, we can think of it as the set of complex lines (which are planes in the real vector space sense) through the origin in \mathbb{C}^{n+1}. Using a Hermitian inner product on \mathbb{C}^{n+1}, it can also be identified with the set of hyperplanes of codimension 1 (i.e., n-dimensional complex linear subspaces) in \mathbb{C}^{n+1}.

Countable-dimensional

This space, called countable-dimensional complex projective space and denoted \mathbb{C}\mathbb{P}^\infty, is defined as the quotient space of the nonzero elements of a countable-dimensional complex vector space (with the standard topology) over \mathbb{C} by the action of \mathbb{C}^* by scalar multiplication.

Others

We can also consider the real projective space corresponding to any topological complex vector space, possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is V, the projective space is defined as follows:we take V \setminus \{ 0 \} with the subspace topology, and then put the quotient topology on its quotient under the action of \mathbb{C}^*.

Particular cases

n Complex projective space \mathbb{C}\mathbb{P}^n
0 one-point space
1 complex projective line, which turns out to be homeomorphic to the 2-sphere
2 complex projective plane
3 link: Fill this in later
countable (\infty) countable-dimensional complex projective space

Algebraic topology

Homology

Further information: homology of complex projective space

Cohomology

Further information: cohomology of complex projective space

Homotopy

Further information: homotopy of complex projective space