Editing Complex projective space (section)

=== CW-decomposition ===
One useful way to construct the complex projective spaces <math>\mathbf{CP}^n</math> is through a recursive construction using [[CW complex|CW-complexes]]. Recall that there is a homeomorphism <math>\mathbf{CP}^1 \cong S^2</math> to the 2-sphere, giving the first space. We can then induct on the cells to get a [[Pushout (category theory)|pushout map]] <math display="block">\begin{matrix}
S^3 & \hookrightarrow & D^4 \\
\downarrow & & \downarrow \\
\mathbf{CP}^1 & \to & \mathbf{CP}^2
\end{matrix}</math> where <math>D^4</math> is the four ball, and <math>S^3 \to \mathbf{CP}^1</math> represents the generator in <math>\pi_3(S^2)</math> (hence it is homotopy equivalent to the [[Hopf fibration|Hopf map]]). We can then inductively construct the spaces as pushout diagrams <math display="block">\begin{matrix}
S^{2n-1} & \hookrightarrow & D^{2n} \\
\downarrow & & \downarrow \\
\mathbf{CP}^{n-1} & \to & \mathbf{CP}^n
\end{matrix}</math> where <math>S^{2n-1} \to \mathbf{CP}^{n-1}</math> represents an element in <math display="block">\begin{align}
\pi_{2n-1}(\mathbf{CP}^{n-1}) &\cong \pi_{2n-1}(S^{2n-2}) \\
&\cong \mathbb{Z}/2
\end{align}</math> The isomorphism of homotopy groups is described below, and the isomorphism of homotopy groups is a standard calculation in [[stable homotopy theory]] (which can be done with the [[Serre spectral sequence]], [[Freudenthal suspension theorem]], and the [[Postnikov tower]]). The map comes from the [[fiber bundle]] <math display="block">S^1 \hookrightarrow S^{2n-1} \twoheadrightarrow \mathbf{CP}^{n-1}</math> giving a non-contractible map, hence it represents the generator in <math>\mathbb{Z}/2</math>. Otherwise, there would be a homotopy equivalence <math>\mathbf{CP}^n \simeq \mathbf{CP}^{n-1}\times D^n</math>, but then it would be homotopy equivalent to <math>S^2</math>, a contradiction which can be seen by looking at the homotopy groups of the space.