Editing Complex projective space (section)

==Topology==
The topology of '''CP'''<sup>''n''</sup> is determined inductively by the following [[CW complex|cell decomposition]].  Let ''H'' be a fixed hyperplane through the origin in '''C'''<sup>''n''+1</sup>.  Under the projection map {{nowrap|'''C'''<sup>''n''+1</sup>\{0} &rarr; '''CP'''<sup>''n''</sup>}}, ''H'' goes into a subspace that is homeomorphic to '''CP'''<sup>''n''&minus;1</sup>.  The complement of the image of ''H'' in '''CP'''<sup>''n''</sup> is homeomorphic to '''C'''<sup>''n''</sup>.  Thus '''CP'''<sup>''n''</sup> arises by attaching a 2''n''-cell to '''CP'''<sup>''n''&minus;1</sup>:
:<math>\mathbf{CP}^n = \mathbf{CP}^{n-1}\cup \mathbf{C}^n.</math>
Alternatively, if the 2''n''-cell is regarded instead as the open unit ball in '''C'''<sup>''n''</sup>, then the attaching map is the Hopf fibration of the boundary.  An analogous inductive cell decomposition is true for all of the projective spaces; see {{harv|Besse|1978}}.

=== 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.

===Point-set topology===
Complex projective space is [[compact space|compact]] and [[connected space|connected]], being a quotient of a compact, connected space.

===Homotopy groups===

From the fiber bundle
:<math>S^1 \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbf{CP}^n</math>
or more suggestively
:<math>U(1) \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbf{CP}^n</math>
'''CP'''<sup>''n''</sup> is [[simply connected]].  Moreover, by the [[long exact homotopy sequence]], the second homotopy group is {{nowrap|1=π<sub>2</sub>('''CP'''<sup>''n''</sup>) ≅ '''Z'''}}, and all the higher homotopy groups agree with those of ''S''<sup>2''n''+1</sup>: {{nowrap|1=π<sub>''k''</sub>('''CP'''<sup>''n''</sup>) ≅ π<sub>''k''</sub>(''S''<sup>2''n''+1</sup>)}} for all ''k'' > 2.

===Homology===
In general, the [[algebraic topology]] of '''CP'''<sup>''n''</sup> is based on the rank of the [[homology group]]s being zero in odd dimensions; also ''H''<sub>2''i''</sub>('''CP'''<sup>''n''</sup>, '''Z''') is [[infinite cyclic]] for ''i'' = 0 to ''n''. Therefore, the [[Betti number]]s run
:1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...
That is, 0 in odd dimensions, 1 in even dimensions 0 through 2n. The [[Euler characteristic]] of '''CP'''<sup>''n''</sup> is therefore ''n''&nbsp;+&nbsp;1. By [[Poincaré duality]] the same is true for the ranks of the [[cohomology group]]s. In the case of cohomology, one can go further, and identify the [[graded ring]] structure, for [[cup product]]; the generator of ''H''<sup>2</sup>('''CP'''<sup>n</sup>, '''Z''') is the class associated to a [[hyperplane]], and this is a ring generator, so that the ring is isomorphic with
:'''Z'''[''T'']/(''T''<sup>''n''+1</sup>),

with ''T'' a degree two generator. This implies also that the [[Hodge number]] ''h''<sup>''i'',''i''</sup> = 1, and all the others are zero.  See {{harv|Besse|1978}}.

===''K''-theory===
It follows from induction and [[Bott periodicity]] that

:<math>K_\mathbf{C}^*(\mathbf{CP}^n) = K_\mathbf{C}^0(\mathbf{CP}^n) = \mathbf{Z}[H]/(H-1)^{n+1}.</math>

The [[tangent bundle]] satisfies

:<math>T\mathbf{CP}^n \oplus \vartheta^1 = H^{\oplus n+1},</math>

where <math>\vartheta^1</math> denotes the trivial line bundle, from the [[Euler sequence]]. From this, the [[Chern class]]es and [[characteristic number]]s can be calculated explicitly.

===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}}.