\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{complex projective space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{cell_structure}{Cell structure}\dotfill \pageref*{cell_structure} \linebreak
\noindent\hyperlink{homotopy}{Homotopy}\dotfill \pageref*{homotopy} \linebreak
\noindent\hyperlink{Cohomology}{Homology and Cohomology}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{ordinary}{Ordinary}\dotfill \pageref*{ordinary} \linebreak
\noindent\hyperlink{complexoriented}{Complex-oriented}\dotfill \pageref*{complexoriented} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Complex projective space $\mathbb{C}P^n$ is the [[projective space]] $\mathbb{A}P^n$ for $\mathbb{A} = \mathbb{C}$ being the [[complex numbers]] (and for $n \in \mathbb{N}$), a [[complex manifold]] of complex [[dimension]] $n$ (real dimension $2n$). Equivalently, this is the complex [[Grassmannian]] $Gr_1(\mathbb{C}^{n+1})$. For the special case $n = 1$ then $\mathbb{C}P^1 \simeq S^2$ is the [[Riemann sphere]].

As $n$ ranges, there are natural inclusions

\begin{displaymath}
\ast = \mathbb{C}P^0
    \hookrightarrow
  \mathbb{C}P^1
    \hookrightarrow
  \mathbb{C}P^2
    \hookrightarrow
  \mathbb{C}P^3
    \hookrightarrow
  \cdots
  \,.
\end{displaymath}
The [[sequential colimit]] over this sequence is the infinite complex projective space $\mathbb{C}P^\infty$. This is a model for the [[classifying space]] $B U(1)$ of [[circle principal bundles]]/[[complex line bundles]] (an [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{ComplexProjectiveSpace}\hypertarget{ComplexProjectiveSpace}{}
For $n \in \mathbb{N}$, then \textbf{complex $n$-dimensional complex projective space} is the [[complex manifold]] (often just regarded as its underlying [[topological space]]) defined as the [[quotient]]

\begin{displaymath}
\mathbb{C}P^n \coloneqq (\mathbb{C}^{n+1}-\{0\})/_\sim
\end{displaymath}
of the [[Cartesian product]] of $(n+1)$-copies of the [[complex plane]], with the origin removed, by the [[equivalence relation]]

\begin{displaymath}
(z \sim w) \Leftrightarrow (z = \kappa \cdot w)
\end{displaymath}
for some $\kappa \in \mathbb{C} - \{0\}$ and using the canonical multiplicative [[action]] of $\mathbb{C}$ on $\mathbb{C}^{n+1}$.

The canonical inclusions

\begin{displaymath}
\mathbb{C}^{n+1} \hookrightarrow \mathbb{C}^{n+2}
\end{displaymath}
induce canonical inclusions

\begin{displaymath}
\mathbb{C}P^n \hookrightarrow \mathbb{C}P^{n+1}
  \,.
\end{displaymath}
The [[sequential colimit]] over this sequence of inclusions is the \emph{infinite complex projective space}

\begin{displaymath}
\mathbb{C}P^\infty
    \coloneqq
  \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n
  \,.
\end{displaymath}
\end{defn}
The following equivalent characterizations are immediate but useful:

\begin{prop}
\label{ComplexProjectiveSpaceAsGrassmannian}\hypertarget{ComplexProjectiveSpaceAsGrassmannian}{}
For $n \in \mathbb{N}$ then complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently the complex [[Grassmannian]]

\begin{displaymath}
\mathbb{C}P^n \simeq Gr_1(\mathbb{C}^{n+1})
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{ComplexProjectiveSpaceAsS1Quotient}\hypertarget{ComplexProjectiveSpaceAsS1Quotient}{}
For $n \in \mathbb{N}$ then complex projective space, def. \ref{ComplexProjectiveSpace}, is equivalently

\begin{enumerate}%
\item the [[coset]]

\begin{displaymath}
\mathbb{C}P^n \simeq U(n+1)/(U(n) \times U(1))
  \,,
\end{displaymath}

\item the quotient of the [[n-sphere|(2n+1)-sphere]] by the [[circle group]] $S^1 \simeq \{ \kappa  \in \mathbb{C}| {\vert \kappa \vert} = 1\}$



\end{enumerate}
\begin{displaymath}
\mathbb{C}P^n \simeq S^{2n+1}/S^1
    \,.
\end{displaymath}
\end{prop}
\begin{proof}
To see the second characterization from def. \ref{ComplexProjectiveSpace}:

With ${\vert -\vert} \colon \mathbb{C}^{n} \longrightarrow \mathbb{R}$ the standard [[norm]], we have that every element $\vec z \in \mathbb{C}^{n+1}$ is identified under the defining equivalence relation with

\begin{displaymath}
\frac{1}{\vert \vec z\vert}\vec z \in S^{2n+1} \hookrightarrow \mathbb{C}^{n+1}
\end{displaymath}
lying on the unit $(2n+1)$-sphere. This fixes the action of $\mathbb{C}-0$ up to a remaining action of complex numbers of unit [[absolute value]]. These form the [[circle group]] $S^1$. This shows that we have a [[commuting diagram]] of functions of underlying sets of the form

\begin{displaymath}
\itexarray{
    S^{2n+1} &\hookrightarrow& \mathbb{C}^{n+1} \setminus \{0\}
    \\
    {}^{\mathllap{q_{S^{2n+1}}}}\downarrow &\searrow^{\mathrlap{f}}& \downarrow^{\mathrlap{q_{\mathbb{C}^{n+1}}}}
    \\
    S^{2n+1}/S^1 &\longrightarrow& \mathbb{C}P^n
  }
\end{displaymath}
where the top horizontal and the two vertical functions are [[continuous function|continuous]], and where the bottom function is a [[bijection]]. Since the diagonal [[composition|composite]] is also continuous, the nature of the [[quotient space topology]] implies that the bottom function is also continuous. To see that it is a [[homeomorphism]] it hence remains to see that it is an [[open map]] (by \href{homeomorphism#HomeoContinuousOpenBijection}{this prop.}).

So let $U \subset S^{2n+1}/S^1$ be an open set, which means that $q_{S^{2n+1}}^{-1}(U) \subset S^{2n+1}$ is an open set. We need to see that $f(q_{S^{2n+1}}^{-1}(U)) \subset \mathbb{C}P^{n}$ is open, hence that $q_{\mathbb{C}^{n+1}}^{-1}(f(q_{S^{2n+1}}^{-1}(U))) \subset \mathbb{C}^{n+1}$ is open. Now by the nature of the [[Euclidean space|Euclidean]] [[metric topology]], the open subset $q_{S^{2n+1}}^{-1}(U)$ is a union of open balls $B^\circ_x(\epsilon)$ in $\mathbb{C}^{n+1}$ intersected with $S^{2n+1}$. But then $q_{\mathbb{C}^{n+1}}^{-1}(f(B^\circ_x(\epsilon)\vert_{S^{2n+1}}))$ is their [[orbit]] under the multiplicative action by $\mathbb{C} \setminus \{0\}$, hence is a [[cylinder]] $B^\circ_x(\epsilon)\vert_{S^{2n+1}} \times (\mathbb{C} \setminus \{0\})$. This is clearly open.

The first characterization follows via prop. \ref{ComplexProjectiveSpaceAsGrassmannian} from the general discusion at \emph{[[Grassmannian]]}. With this the second characterization follows also with the [[coset]] identification of the $(2n+1)$-sphere: $S^{2n+1} \simeq U(n+1)/U(n)$ (\href{unitary+group#nSphereAsUnitaryCosetSpace}{exmpl.}).

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[complex projective curve]]


\item [[complex projective plane]]



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\hypertarget{cell_structure}{}\subsubsection*{{Cell structure}}\label{cell_structure}

\begin{prop}
\label{CellComplexStructureOnComplexProjectiveSpace}\hypertarget{CellComplexStructureOnComplexProjectiveSpace}{}
There is a [[CW-complex]] structure on complex projective space $\mathbb{C}P^n$ (def. \ref{ComplexProjectiveSpace}) for $n \in \mathbb{N}$, given by [[induction]], where $\mathbb{C}P^{n+1}$ arises from $\mathbb{C}P^n$ by [[attaching space|attaching]] a single cell of dimension $2(n+1)$ with [[attaching map]] the [[projection]] $S^{2n+1} \longrightarrow \mathbb{C}P^n$ from prop. \ref{ComplexProjectiveSpaceAsS1Quotient}:

\begin{displaymath}
\itexarray{
     S^{2n+1} &\longrightarrow& S^{2n+1}/S^1 \simeq \mathbb{C}P^n
     \\
     {}^{\mathllap{\iota_{2n+2}}}\downarrow^{\mathrlap{i_n}}
       &(po)& \downarrow
     \\
     D^{2n+2} &\underset{q}{\longrightarrow}& \mathbb{C}P^{n+1}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Given [[homogeneous coordinates]] $(z_0 ,  z_1 ,  \cdots ,  z_n ,  z_{n+1} ,  z_{n+2}) \in \mathbb{C}^{n+2}$ for $\mathbb{C}P^{n+1}$, let

\begin{displaymath}
\phi \coloneqq -arg(z_{n+2})
\end{displaymath}
be the [[phase]] of $z_{n+2}$. Then under the equivalence relation defining $\mathbb{C}P^{n+1}$ these coordinates represent the same element as

\begin{displaymath}
\frac{1}{\vert \vec z\vert}(e^{i \phi} z_0, e^{i \phi}z_1,\cdots, e^{i \phi}z_{n+1}, r)
  \,,
\end{displaymath}
where

\begin{displaymath}
r = {\vert z_{n+2}\vert}\in [0,1) \subset \mathbb{C}
\end{displaymath}
is the [[absolute value]] of $z_{n+2}$. Representatives $\vec z'$ of this form (${\vert \vec z' \vert = 1}$ and $z'_{n+2} \in [0,1]$) parameterize the [[n-disk|2n+2-disk]] $D^{2n+2}$ with [[boundary]] the $(2n+1)$-sphere at $r = 0$.

The resulting function $q \colon D^{2n+2} \to \mathbb{C}P^{n+1}$ is continuous: It may be factored as

\begin{displaymath}
\itexarray{
    q_{D^{2n+2}} \colon & D^{2n+2} &\overset{\phantom{AAA}}{\hookrightarrow}& \mathbb{C}^{n+2} \setminus \{0\} &\overset{q_{\mathbb{C}^{n+2}}}{\longrightarrow}&  \mathbb{C}P^{n+1}
    \\
    & (Re(z_1), Im(z_1),  \cdots, Re(z_{n+1}), Im(z_{n+1}), r) &\mapsto& (z_1, \cdots, z_{n+1}, r)  &\mapsto& [ z_1 : \cdots : z_{n+1} : r ]
  }
\end{displaymath}
and here the first map is the [[embedding of topological spaces|embedding]] of the disk $D^{2n+2}$ as a [[hemisphere]] in $\mathbb{R}^{2n+1} \hookrightarow \mathbb{R}^{2n+2} \simeq \mathbb{C}^{2n+2}$, while the second is the defining quotient space projection. Both of these are continuous, and hence so is their composite.

The only remaining part of the action of $\mathbb{C}-\{0\}$ which fixes the conditions ${\vert z'\vert} = 0$ and $z'_{n+2}$ is $S^1 \subset \mathbb{C} \setminus \{0\}$ acting on the elements with $r = \{z'_{n+2}\} = 0$ by phase shifts on the $z_0, \cdots, z_{n+1}$. The quotient of this remaining action on $D^{2(n+1)}$ identifies its boundary $S^{2n+1}$-sphere with $\mathbb{C}P^{n}$, by prop. \ref{ComplexProjectiveSpaceAsS1Quotient}.

This shows that the above square is a [[pushout]] diagram of underlying sets.

By the nature of [[colimits]] in [[Top]] (\href{Top#DescriptionOfLimitsAndColimitsInTop}{this prop.}) it remains to see that the [[topological space|topology]] on $\mathbb{C}P^{n+1}$ is the [[final topology]] induced by the functions $D^{2n+2} \to \mathbb{C}P^{n+1}$ and $\mathbb{C}P^n \to \mathbb{C}P^{n+1}$, hence that a subset of $\mathbb{C}P^{n+1}$ is open precisely if its pre-images under these two functions are open.

We saw above that $q_{D^{2n+2}}$ is continuous. Moreover, also the function $i_n \colon \mathbb{C}P^n \to \mathbb{C}P^{n+1}$ is continuous (by \href{projective+space#CanonicalInclusionOfProjectiveSpaces}{this lemma}).

This shows that if a subset of $\mathbb{C}P^{n+1}$ is open, then its pre-images under these functions are open. It remains to see that if $S \subset \mathbb{C}P^{n+1}$ is a subset with $q_{S^{2n+2}}^{-1}(S) \subset D^{2n+2}$ open and $i_n^{-1}(S) \subset \mathbb{C}P^n$ open, then $S \subset \mathbb{C}P^{n+1}$ is open.

Notice that $q_{\mathbb{C}^{n+2}}^{-1}(S)$ contains with every point also its [[orbit]] under the [[action]] of $\mathbb{C} \setminus \{0\}$, and that every open subset of $D^{2n+2}$ is a unions of open balls. By the above factorization of $q_{D^{2n+2}}$ this means that if $q_{D^{2n+2}}^{-1}(S)$ is open, then $q_{\mathbb{C}^{n+2}}^{-1}(S)$ is a union of open cyclinders, hence is open. By the nature of the [[quotient topology]], this means that $S \subset \mathbb{C}P^n$ is open.

\end{proof}
\hypertarget{homotopy}{}\subsubsection*{{Homotopy}}\label{homotopy}

Write $\Sigma^\infty (\mathbb{C}P^\infty)_+ \in Ho(Spectra)$ for the [[H-group ring spectrum]] of $\mathbb{C}P^\infty \simeq B U(1)$ (see there for details).

For $X \in Ho(Top)$ the [[homotopy type]] of a [[topological space]] in the [[classical homotopy category]], write

\begin{displaymath}
[\Sigma^\infty X_+ , \Sigma^\infty, \mathbb{C}P^\infty_+]
   \simeq
  [X, \Omega^\infty \Sigma^\infty \mathbb{C}P^\infty_+]
  \in Ab
\end{displaymath}
for the [[hom-group]] in the [[stable homotopy category]], which, by [[adjunction]], is equivalently computed in the [[classical homotopy category]] as shown on the right.

Write

\begin{displaymath}
i \;\colon\; \mathbb{C}P^\infty \simeq B U(1) \simeq B U(1) \times \{1\}
 \hookrightarrow B U \times \mathbb{Z}
\end{displaymath}
for the inclusion into the [[classifying space]] for complex [[topological K-theory]] which classifies the inlusion of [[complex line bundles]] $E$ as [[virtual vector bundles]] $[E] - 0$.

\begin{prop}
\label{HGroupRingSpectrumSurjectsOntoTopologicalKTheory}\hypertarget{HGroupRingSpectrumSurjectsOntoTopologicalKTheory}{}
For any $X \in Ho(Top)$ the [[ring homomorphism]]

\begin{displaymath}
i_\ast \;\colon\;
  [\Sigma^\infty X_+, \Sigma^\infty \mathbb{C}P_+]
    \longrightarrow
  K_{\mathbb{C}}(X)
\end{displaymath}
to [[topological K-theory]] is [[surjective function|surjective]].

\end{prop}
This is due to (\hyperlink{Segal73}{Segal 73, prop. 1}).

Prop. \ref{HGroupRingSpectrumSurjectsOntoTopologicalKTheory} is sharpened by \emph{[[Snaith's theorem]]}. See there for more. The version for [[real projective space]] is called the \emph{[[Kahn-Priddy theorem]]}. \#

\hypertarget{Cohomology}{}\subsubsection*{{Homology and Cohomology}}\label{Cohomology}

\hypertarget{ordinary}{}\paragraph*{{Ordinary}}\label{ordinary}

\begin{prop}
\label{OrdinaryCohomologyOfComplexProjectiveSpace}\hypertarget{OrdinaryCohomologyOfComplexProjectiveSpace}{}
For $A \in$ [[Ab]] any [[abelian group]], then the [[ordinary homology]] [[homology groups|groups]] of complex projective space $\mathbb{C}P^n$ with [[coefficients]] in $A$ are

\begin{displaymath}
H_k(\mathbb{C}P^n,A)\simeq
  \left\{
    \itexarray{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Similarly the [[ordinary cohomology]] [[cohomology groups|groups]] of $\mathbb{C}P^n$ is

\begin{displaymath}
H^k(\mathbb{C}P^n,A)
   \simeq
  \left\{
    \itexarray{
       A & for \; k \;even\; and \; k \leq 2n
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Moreover, if $A$ carries the structure of a [[ring]] $R = (A, \cdot)$, then under the [[cup product]] the [[cohomology ring]] of $\mathbb{C}P^n$ is the the [[graded ring]]

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, R)
   \simeq
  R[c_1] / (c_1^{n+1})
\end{displaymath}
which is the [[quotient]] of the [[polynomial ring]] on a single generator $c_1$ in degree 2, by the relation that identifies [[cup products]] of more than $n$-copies of the generator $c_1$ with zero.

Finally, the [[cohomology ring]] of the infinite-dimensional complex projective space is the [[formal power series ring]] in one generator:

\begin{displaymath}
H^\bullet(\mathbb{C}P^\infty, R)
   \simeq
  R[ [ c_1 ] ]
  \,.
\end{displaymath}
(Or else the [[polynomial ring]] $R[c_1]$, depending on how one chooses to extract a ring from a graded ring, see remark \ref{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}.)

\end{prop}
\begin{proof}
First consider the case that the coefficients are the [[integers]] $A = \mathbb{Z}$.

Since $\mathbb{C}P^n$ admits the structure of a [[CW-complex]] by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, we may compute its [[ordinary homology]] equivalently as its [[cellular homology]] (\href{Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence}{thm.}). By definition (\href{cellular+homology#CellularChainComplex}{defn.}) this is the [[chain homology]] of the chain complex of [[relative homology]] groups

\begin{displaymath}
\cdots
    \overset{\partial_{cell}}{\longrightarrow}
  H_{q+2}((\mathbb{C}P^n)_{q+2}, (\mathbb{C}P^n)_{q+1})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q+1}((\mathbb{C}P^n)_{q+1}, (\mathbb{C}P^n)_{q})
   \overset{\partial_{cell}}{\longrightarrow}
  H_{q}((\mathbb{C}P^n)_{q}, (\mathbb{C}P^n)_{q-1})
    \overset{\partial_{cell}}{\longrightarrow}
  \cdots
  \,,
\end{displaymath}
where $(-)_q$ denotes the $q$th stage of the [[CW-complex]]-structure. Using the CW-complex structure provided by prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, then there are cells only in every second degree, so that

\begin{displaymath}
(\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P)_{2k}
\end{displaymath}
for all $k \in \mathbb{N}$. It follows that the cellular chain complex has a zero group in every second degree, so that all differentials vanish. Finally, since prop. \ref{CellComplexStructureOnComplexProjectiveSpace} says that $(\mathbb{C}P^n)_{2k+2}$ arises from $(\mathbb{C}P^n)_{2k+1} = (\mathbb{C}P^n)_{2k}$ by attaching a single $2k+2$-cell it follows that (by passage to [[reduced homology]])

\begin{displaymath}
H_{2k}(\mathbb{C}P^n, \mathbb{Z})
    \simeq
  \tilde H_{2k}(S^{2k})((\mathbb{C}P^n)_{2k}/(\mathbb{C}P^n)_{2k-1})
   \simeq
  \tilde H_{2k}(S^{2k})
    \simeq
  \mathbb{Z}
  \,.
\end{displaymath}
This establishes the claim for ordinary homology with integer coefficients.

In particular this means that $H_q(\mathbb{C}P^n, \mathbb{Z})$ is a [[free abelian group]] for all $q$. Since free abelian groups are the [[projective objects]] in [[Ab]] (\href{projective+object#ProjectiveObjectsInAbAreFreeGroups}{prop.}) it follows (with the discussion at \emph{[[derived functors in homological algebra]]}) that the [[Ext]]-groups vanishe:

\begin{displaymath}
Ext^1(H_q(\mathbb{C}P^n, \mathbb{Z}),A) = 0
\end{displaymath}
and the [[Tor]]-groups vanishes:

\begin{displaymath}
Tor_1(H_q(\mathbb{C}P^n), A) = 0
  \,.
\end{displaymath}
With this, the statement about homology and cohomology groups with general coefficients follows with the [[universal coefficient theorem]] for ordinary homology (\href{universal+coefficient+theorem#TheoremInOrdinaryHomology}{thm.}) and for ordinary cohomology (\href{universal+coefficient+theorem#OrdinaryStatementInCohomology}{thm.}).

Finally to see the action of the [[cup product]]: by definition this is the composite

\begin{displaymath}
\cup
  \;\colon\;
  H^\bullet(\mathbb{C}P^n, R)
  \otimes
  H^\bullet(\mathbb{C}P^n, R)
    \longrightarrow
  H^\bullet(\mathbb{C}P^n \times \mathbb{C}P^n , R)
    \overset{\Delta^\ast}{\longrightarrow}
  H^\bullet(\mathbb{C}P^n,R)
\end{displaymath}
of the ``cross-product'' map that appears in the [[Kunneth theorem]], and the pullback along the [[diagonal]] $\Delta\colon \mathbb{C}P^n \to \mathbb{C}P^n \times \mathbb{C}P^n$.

Since, by the above, the groups $H^{2k}(\mathbb{C}P^n,R) \simeq R[2k]$ and $H^{2k+1}(\mathbb{C}P^n,R) = 0$ are free and finitely generated, the [[Kunneth theorem]] in ordinary cohomology applies (\href{K&#252;nneth+theorem#KunnethInOrdinaryCohomology}{prop.}) and says that the cross-product map above is an isomorphism. This shows that under cup product pairs of generators are sent to a generator, and so the statement $H^\bullet(\mathbb{C}P^n , R)\simeq R[c_1](c_1^{n+1})$ follows.

This also implies that the projection maps

\begin{displaymath}
H^\bullet((\mathbb{C}P^\infty)_{2n+2}, R)
  =
  H^\bullet(\mathbb{C}P^{n+1}, R)
  \to
  H^\bullet(\mathbb{C}P^{n+}, R)
  =
  H^\bullet((\mathbb{C}P^\infty)_{2n}, R)
\end{displaymath}
are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] (\href{lim1#MittagLefflerCondition}{def.}, \href{lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections}{exmpl.}) and therefore the [[Milnor exact sequence]] for cohomology (\href{lim1#MilorSequenceForReducedCohomologyOnCWComplex}{prop.}) implies the last claim to be proven:

\begin{displaymath}
\begin{aligned}
    H^\bullet(\mathbb{C}P^\infty, R)
    \\
    &
     \simeq
    H^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n , R)
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n H^\bullet(\mathbb{C}P^n, R)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( R [c_1^E] / ((c_1)^{n+1})  )
   \\
   & \simeq R[ [ c_1 ] ]
   \,,
  \end{aligned}
\end{displaymath}
where the last step is \href{formal+scheme#FormalPowerSeries}{this prop.}.

\end{proof}
\begin{remark}
\label{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}\hypertarget{ChoiceOfRingStructureOnGradedCohomologyGroupOfMultiplicativeCohomologyTheory}{}
There is in general a choice to be made in interpreting the [[cohomology groups]] of a [[multiplicative cohomology theory]] $E$ as a [[ring]]:

a priori $E^\bullet(X)$ is a sequence

\begin{displaymath}
\{E^n(X)\}_{n \in \mathbb{Z}}
\end{displaymath}
of [[abelian groups]], together with a system of group homomorphisms

\begin{displaymath}
E^{n_1}(X) \otimes E^{n_2}(X)
    \longrightarrow
  E^{n_1 + n_2}(X)
  \,,
\end{displaymath}
one for each pair $(n_1,n_2) \in \mathbb{Z}\times\mathbb{Z}$.

In turning this into a single [[ring]] by forming [[formal sums]] of elements in the groups $E^n(X)$, there is in general the choice of whether allowing formal sums of only finitely many elements, or allowing arbitrary formal sums.

In the former case the ring obtained is the [[direct sum]]

\begin{displaymath}
\oplus_{n \in \mathbb{N}} E^n(X)
\end{displaymath}
while in the latter case it is the [[Cartesian product]]

\begin{displaymath}
\prod_{n \in \mathbb{N}} E^n (X)
  \,.
\end{displaymath}
These differ in general. For instance if $E$ is [[ordinary cohomology]] with [[integer]] [[coefficients]] and $X$ is complex projective space $\mathbb{C}P^\infty$, then (prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace})

\begin{displaymath}
E^n(X)
  =
  \left\{
    \itexarray{
      \mathbb{Z} & n \; even
      \\
      0 & otherwise
    }
  \right.
\end{displaymath}
and the product operation is given by

\begin{displaymath}
E^{2{n_1}}(X)\otimes E^{2 n_2}(X)
    \longrightarrow
  E^{2(n_1 + n_2)}(X)
\end{displaymath}
for all $n_1, n_2$ (and zero in odd degrees, necessarily). Now taking the [[direct sum]] of these, this is the [[polynomial ring]] on one generator (in degree 2)

\begin{displaymath}
\oplus_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z}[c_1]
  \,.
\end{displaymath}
But taking the [[Cartesian product]], then this is the [[formal power series ring]]

\begin{displaymath}
\prod_{n \in \mathbb{N}} E^n(X) \;\simeq\; \mathbb{Z} [ [ c_1 ] ]
  \,.
\end{displaymath}
A priori both of these are sensible choices. The former is the usual choice in traditional [[algebraic topology]]. However, from the point of view of regarding [[ordinary cohomology]] theory as a [[multiplicative cohomology theory]] right away, then the second perspective tends to be more natural:

The cohomology of $\mathbb{C}P^\infty$ is naturally computed as the [[inverse limit]] of the cohomolgies of the $\mathbb{C}P^n$, each of which unambiguously has the ring structure $\mathbb{Z}[c_1]/((c_1)^{n+1})$. So we may naturally take the limit in the [[category]] of [[commutative rings]] right away, instead of first taking it in $\mathbb{Z}$-indexed sequences of abelian groups, and then looking for ring structure on the result. But the limit taken in the category of rings gives the [[formal power series ring]] (see \href{formal+scheme#FormalPowerSeries}{here}).

Incidentally, this is the default choice of ring structure for [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[multiplicative cohomology theories]] evaluated on $\mathbb{C}P^\infty$. In particular in [[complex oriented cohomology]] (see there) this choice is of paramount importance.

See also for instance remark 1.1. in [[Jacob Lurie]]: \emph{[[A Survey of Elliptic Cohomology]]}.

\end{remark}
\hypertarget{complexoriented}{}\paragraph*{{Complex-oriented}}\label{complexoriented}

\begin{prop}
\label{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}\hypertarget{CohomologyRingOfBU1ForComplexOrientedCohomologyTheory}{}
Given a [[complex oriented cohomology theory]] $(E^\bullet, c^E_1)$ (\href{complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheory}{defn.}), then there is an [[isomorphism]] of [[graded rings]]

\begin{displaymath}
E^\bullet(\mathbb{C}P^\infty) \simeq E^\bullet(\ast)[ [ c_1^E ] ]
\end{displaymath}
between the $E$-[[cohomology ring]] of infinite-dimensional complex projective space and the [[formal power series]] in one generator of even degree over the $E$-[[cohomology ring]] of the point.

\end{prop}
\begin{proof}
Using the [[CW-complex]]-structure on $\mathbb{C}P^\infty$ from prop. \ref{CellComplexStructureOnComplexProjectiveSpace}, given by inductively identifying $\mathbb{C}P^{n+1}$ with the result of attaching a single $2n$-cell to $\mathbb{C}P^n$. With this structure, the unique 2-cell inclusion $i \;\colon\; S^2 \hookrightarrow \mathbb{C}P^\infty$ is identified with the canonical map $S^2 \to B U(1)$.

Then consider the [[Atiyah-Hirzebruch spectral sequence]] for the $E$-cohomology of $\mathbb{C}P^n$.

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \;\Rightarrow\;
  E^\bullet(\mathbb{C}P^n)
  \,.
\end{displaymath}
Since, by prop. \ref{OrdinaryCohomologyOfComplexProjectiveSpace}, the [[ordinary cohomology]] with [[integer]] [[coefficients]] of projective space is

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[c_1]/((c_1)^{n+1})
  \,,
\end{displaymath}
where $c_1$ represents a unit in $H^2(S^2, \mathbb{Z})\simeq \mathbb{Z}$, and since similarly the [[ordinary homology]] of $\mathbb{C}P^n$ is a [[free abelian group]], hence a [[projective object]] in abelian groups (\href{projective+object#ProjectiveObjectsInAbAreFreeGroups}{prop.}), the [[Ext]]-group vanishes in each degree ($Ext^1(H_n(\mathbb{C}P^n), E^\bullet(\ast)) = 0$) and so the [[universal coefficient theorem]] (\href{universal+coefficient+theorem#OrdinaryStatementInCohomology}{prop.}) gives that the second page of the spectral sequence is

\begin{displaymath}
H^\bullet(\mathbb{C}P^n, E^\bullet(\ast))
  \simeq
  E^\bullet(\ast)[ c_1 ] / (c_1^{n+1})
  \,.
\end{displaymath}
By the standard construction of the [[Atiyah-Hirzebruch spectral sequence]] (\href{Atiyah&#8211;Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace}{here}) in this identification the element $c_1$ is identified with a generator of the [[relative cohomology]]

\begin{displaymath}
E^2((\mathbb{C}P^n)_2, (\mathbb{C}P^n)_1)
  \simeq
  \tilde E^2(S^2)
\end{displaymath}
(using, by the above, that this $S^2$ is the unique 2-cell of $\mathbb{C}P^n$ in the standard cell model).

This means that $c_1$ is a permanent cocycle of the spectral sequence (in the kernel of all differentials) precisely if it arises via restriction from an element in $E^2(\mathbb{C}P^n)$ and hence precisely if there exists a complex orientation $c_1^E$ on $E$. Since this is the case by assumption on $E$, $c_1$ is a permanent cocycle. (For the fully detailed argument see (\hyperlink{Pedrotti16}{Pedrotti 16}).)

The same argument applied to all elements in $E^\bullet(\ast)[c]$, or else the $E^\bullet(\ast)$-linearity of the differentials (\href{multiplicative+cohomology+theory#LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory}{prop.}), implies that all these elements are permanent cocycles.

Since the AHSS of a multiplicative cohomology theory is a [[multiplicative spectral sequence]] (\href{multiplicative+cohomology+theory#AHSSForMultiplicativeCohomologyTheoryIsMultiplicative}{prop.}) this implies that the differentials in fact vanish on all elements of $E^\bullet(\ast) [c_1] / (c_1^{n+1})$, hence that the given AHSS collapses on the second page to give

\begin{displaymath}
\mathcal{E}_\infty^{\bullet,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1^{E} ] / ((c_1^E)^{n+1})
\end{displaymath}
or in more detail:

\begin{displaymath}
\mathcal{E}_\infty^{p,\bullet}
  \simeq
  \left\{
    \itexarray{
      E^\bullet(\ast) & \text{if}\; p \leq  2n \; and\; even
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Moreover, since therefore all $\mathcal{E}_\infty^{p,\bullet}$ are [[free modules]] over $E^\bullet(\ast)$, and since the filter stage inclusions $F^{p+1} E^\bullet(X) \hookrightarrow F^{p}E^\bullet(X)$ are $E^\bullet(\ast)$-[[module]] homomorphisms (\href{multiplicative+cohomology+theory#RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}{prop.}) the \href{spectral+sequence#ExtensionProblem}{extension problem} trivializes, in that all the [[short exact sequences]]

\begin{displaymath}
0 \to F^{p+1}E^{p+\bullet}(X) \longrightarrow F^{p}E^{p+\bullet}(X) \longrightarrow \mathcal{E}_\infty^{p,\bullet} \to 0
\end{displaymath}
[[split exact sequence|split]] (since the [[Ext]]-group $Ext^1_{E^\bullet(\ast)}(\mathcal{E}_\infty^{p,\bullet},-) = 0$ vanishes on the [[free module]], hence [[projective module]] $\mathcal{E}_\infty^{p,\bullet}$).

In conclusion, this gives an isomorphism of graded rings

\begin{displaymath}
E^\bullet(\mathbb{C}P^n)
  \simeq
  \underset{p}{\oplus} \mathcal{E}_\infty^{p,\bullet}
  \simeq
  E^\bullet(\ast)[ c_1 ] / ((c_1^{E})^{n+1})
  \,.
\end{displaymath}
A first consequence is that the projection maps

\begin{displaymath}
E^\bullet((\mathbb{C}P^\infty)_{2n+2})
  =
  E^\bullet(\mathbb{C}P^{n+1})
  \to
  E^\bullet(\mathbb{C}P^{n+})
  =
  E^\bullet((\mathbb{C}P^\infty)_{2n})
\end{displaymath}
are all [[epimorphisms]]. Therefore this sequence satisfies the [[Mittag-Leffler condition]] (\href{lim1#MittagLefflerCondition}{def.}, \href{lim1#MittagLefflerSatisfiedInParticularForTowerOfSurjections}{exmpl.}) and therefore the [[Milnor exact sequence]] for generalized cohomology (\href{lim1#MilorSequenceForReducedCohomologyOnCWComplex}{prop.}) finally implies the claim:

\begin{displaymath}
\begin{aligned}
    E^\bullet(B U(1))
     & \simeq
    E^\bullet(\mathbb{C}P^\infty)
    \\
    & \simeq
    E^\bullet( \underset{\longleftarrow}{\lim}_n \mathbb{C}P^n )
   \\
    &\simeq \underset{\longrightarrow}{\lim}_n E^\bullet(\mathbb{C}P^n)
   \\
   &\simeq \underset{\longrightarrow}{\lim}_n ( E^\bullet(\ast) [c_1^E] / ((c_1^E)^{n+1})  )
   \\
   & \simeq E^\bullet(\ast)[ [ c_1^E ] ]
   \,,
  \end{aligned}
\end{displaymath}
where the last step is \href{formal+scheme#FormalPowerSeries}{this prop.}.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[real projective space]]


\item [[complex oriented cohomology]]


\item [[classifying space]]


\item [[Fubini-Study metric]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

See also

\begin{itemize}%
\item [[Topospaces]], \emph{\href{http://topospaces.subwiki.org/wiki/Complex_projective_space}{Complex projective space}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org
/wiki/Complex_projective_space}{Complex projective space}}



\end{itemize}
Computation of the [[stable homotopy groups]] of $\mathbb{C}P^\infty$ is due to

\begin{itemize}%
\item R. E. Mosher, \emph{Some stable homotopy of complex projective space}, Topology 7 (1968), 179-193 (\href{https://core.ac.uk/download/pdf/82547916.pdf}{pdf})


\item [[Graeme Segal]], \emph{The stable homotopy of complex of projective space}, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], \href{https://doi.org/10.1093/qmath/24.1.1}{doi:10.1093/qmath/24.1.1})



\end{itemize}
(See also at \emph{[[Snaith's theorem]]}.)

Detailed review of the [[Atiyah-Hirzebruch spectral sequence]] for [[complex oriented cohomology]] is in

\begin{itemize}%
\item [[Riccardo Pedrotti]], \emph{Complex oriented cohomology, generalized orientation and Thom isomorphism}, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])

\end{itemize}
[[!redirects complex projective spaces]]

[[!redirects infinite complex projective space]] [[!redirects infinite complex projective spaces]]



\end{document}