Editing Complex projective space (section)

===Classifying space===
There is a space <math>\mathbf{CP}^\infty</math> which, in a sense, is the [[inductive limit]] of <math>\mathbf{CP}^n</math> as <math>n \to \infty</math>. It is [[BU(1)]], the [[classifying space]] of [[U(1)]], the circle group, in the sense of [[homotopy theory]], and so classifies complex [[line bundle]]s. Equivalently it accounts for the first [[Chern class]]. This can be seen heuristically by looking at the fiber bundle maps <math display="block">S^1 \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbf{CP}^n</math> and <math>n \to \infty</math>. This gives a fiber bundle (called the '''<u>universal circle bundle</u>''') <math display="block">S^1 \hookrightarrow S^\infty  \twoheadrightarrow \mathbf{CP}^\infty</math> constructing this space. Note using the long [[exact sequence]] of homotopy groups, we have <math>\pi_2(\mathbf{CP}^\infty) = \pi_1(S^1)</math> hence <math>\mathbf{CP}^\infty</math> is an [[Eilenberg–MacLane space]], a <math>K(\mathbb{Z},2)</math>. Because of this fact, and [[Brown's representability theorem]], we have the following isomorphism <math display="block">H^2(X;\mathbb{Z}) \cong [X,\mathbf{CP}^\infty]</math> for any nice CW-complex <math>X</math>. Moreover, from the theory of [[Chern class|Chern classes]], every complex line bundle <math>L \to X</math> can be represented as a pullback of the universal line bundle on <math>\mathbf{CP}^\infty</math>, meaning there is a pullback square <math display="block">\begin{matrix}
L & \to & \mathcal{L} \\
\downarrow & &\downarrow \\
X & \to & \mathbf{CP}^\infty
\end{matrix}</math> where <math>\mathcal{L} \to \mathbf{CP}^\infty</math> is the associated vector bundle of the principal <math>U(1)</math>-bundle <math>S^\infty \to \mathbf{CP}^\infty</math>. See, for instance, {{harv|Bott|Tu|1982}} and {{harv|Milnor|Stasheff|1974}}.