Real and Complex projective n-spaces

Real projective n-space

Real projective $n$ space, $\mathbb{R} P^n$ is defined to be the space of all lines through the origin in $\mathbb{R}^{n+1}$ . Each such kind is determined by a non-zero vector in $\mathbb{R}^{n+1}$ , unique up to scalar multiplication, and $\mathbb{R} P^n$ is topologized as the quotient space of $\mathbb{R}^{n+1}-\{0\}$ under the equivalence relation $v \sim \lambda v$ for scalars $\lambda \neq 0$ . We can restrict to vectors of length 1, so $\mathbb{R}P^n$ is also the quotient spaces $\mathbf{S}^n / (v \sim -v)$ , the sphere with antipodal points identified. This is equivalent to saying that $\mathbb{R}P^n$ is the quotient space of a hemisphere $\mathbf{D^n}$ with antipodal points of $\partial D^n$ identified. Since $\partial D^n$ with antipodal points identified is just $\mathbb{R}P^{n-1}$ , we see that $\mathbb{R}P^n$ is obtained from $\mathbb{R}P^{n-1}$ by attaching an $n$ -cell, with the quotient projection $S^{n-1} \to \mathbb{R}P^{n-1}$ as the attaching map. It follows by induction on $n$ that $\mathbb{R}P^n$ has a cell complex structure $e^0 \cup e^1 \cup ... \cup e^n$ with one cell $e^i$ in each dimension $i \leq n$ . Since $\mathbb{R}P^n$ is obtained from $\mathbb{R}P^{n-1}$ by attaching an $n$ -cell, the infinite union $\mathbb{R}P^{\infty}={\bigcup}_n \mathbb{R}P^n$ becomes a Cell Complex with one cell in each dimension. We can view $\mathbb{R}P^{\infty}$ as the space of lines through the origin in $\mathbb{R}^{\infty}={\bigcup}_n \mathbb{R}^n$ .

Complex Projective n-space

Complex projective n-space $\mathbb{C}P^n$ is space of complex lines through the origin in $\mathbb{C}P^{n+1}$ , that is, 1-dimensional vector subspaces of $\mathbb{C}^{n+1}$ . As in the case of Real projective n-space, each line is determined by a non-zero vector in $\mathbb{C}^{n+1}$ , unique up to scalar multiplication, and $\mathbb{C}P^n$ is topologized as the quotient space of $\mathbb{C}^{n+1}-0$ under the equivalence relation $v \sim \lambda v$ for $\lambda = 0$ .

Equivalently, this is the quotient of the unit Sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$ with $v \sim \lambda v$ for $| \lambda |=1$ .

It is also possible to obtain $\mathbb{C}P^n$ as a quotient space of disk $D^{2n}$ under the identifications $v \sim \lambda v$ for $\lambda v \in \partial D^{2n}$ , in the following way:

The vectors in $S^{2n+1} \subset \mathbb{C}^{n+1}$ with last coordinate real and nonnegative are precisely the vectors of the form $(\omega, \sqrt{1-{|\omega|}^2} ) \in \mathbb{C}^n \times \mathbb{C}$ with $|\omega| \leq 1$ . Such vectors form the of the function $\omega \to \sqrt{1-{|\omega|}^2}$ . This is a disk ${D_+}^{2n}$ bounded by the sphere $S^{2n-1} \subset S^{2n+1}$ consisting of vectors $(\omega, 0) \in \mathbb{C}^n \times \mathbb{C}$ with $|\omega|=1$ . Each vector in $S^{2n+1}$ is equivalent under the identifications $v \sim \lambda v$ to a vector in ${D_+}^{2n}$ , and the latter vector is unique if its last coordinate is non-zero. If the last coordinate is zero, we have just the identifications $v \sim \lambda v$ for $v \in S^{2n-1}$ .

From the description of $\mathbb{C}P^n$ as the quotient of ${D_+}^{2n}$ under the identifications $v \sim \lambda v$ for $v \in S^{2n-1}$ , it follows that $\mathbb{C}P^n$ is obtained from $\mathbb{C}P^{n-1}$ by attaching a cell $e^{2n}$ via the quotient map $S^{2n-1} \to \mathbb{C}P^{n-1}$ . So by induction on $n$ , we obtain a cell structure

$\mathbb{C}P^n=e^0 \cup e^2 \cup ...\cup e^{2n}$

with cells only in even dimensions. Similarly, $\mathbb{C}P^{\infty}$ has a cell structure with one cell in each even dimension.

Note: The Sphere $S^n$ has the structure of a cell complex with just two cells, $e^0$ and $e^n$ , the n-cell being attached by the constant map $S^{n-1} \to e^0$ . This is equivalent to regarding $S^n$ as the quotient space $\dfrac{D^n}{\partial D^n}$ .

How to Draw a Cell Complex (or CW Complex)

Let we try to construct a space by following procedure:

Start with a discrete set ${X^0}$ , whose points are regarded as $\mathbf{0}$ -cells.
Inductively, form the $\textbf{n}$ -skeleton $X^n$ from $X^{n-1}$ by attaching $n$ -cells $e^n_{\alpha}$ via maps $\Phi_{\alpha} : S^{n-1} \rightarrow X^{n-1}$ . This means that $X^n$ is the quotient space of the disjoint union $X^{n-1} \, \mathbf{\sqcup_{\alpha}} D^n_{\alpha}$ of $X^{n-1}$ with a collection of n-disks $D^n_\alpha$ under the identifications $x \sim \Phi_{\alpha} (x)$ for $x \in \partial D^n_\alpha$ . Thus as a set

$X^n=X^{n-1} \mathbf{\sqcup_{\alpha}} e^n_\alpha$

where each $e^n_\alpha$ is an open $n$ disk.

One can either stop this inductive process at a finite stage, setting $X=X^n$ for some $n < \infty$ , one can continue indefinitely, setting $X=\bigcup_n X^n$ .

A space $X$ constructed in this way is called a CELL COMPLEX or CW COMPLEX.

Reference:

ALGEBRAIC TOPOLOGY

Allen Hatcher

http://www.math.cornell.edu/~hatcher/#ATI

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MY DIGITAL NOTEBOOK

Real and Complex projective n-spaces

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Real projective n-space

Complex Projective n-space

How to Draw a Cell Complex (or CW Complex)

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Real and Complex projective n-spaces

Start here

Real projective n-space

Complex Projective n-space

How to Draw a Cell Complex (or CW Complex)

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