\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*{Whitehead tower}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - 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{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{whiteheads_construction}{Whitehead's construction}\dotfill \pageref*{whiteheads_construction} \linebreak
\noindent\hyperlink{functorial_constructions}{Functorial constructions}\dotfill \pageref*{functorial_constructions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OfTheOrthogonalGroup}{Whitehead tower of the orthogonal group}\dotfill \pageref*{OfTheOrthogonalGroup} \linebreak
\noindent\hyperlink{whitehead_tower_in_general_toposes}{Whitehead tower in general $(\infty,1)$-toposes}\dotfill \pageref*{whitehead_tower_in_general_toposes} \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}

The \emph{Whitehead tower} of a [[pointed object|pointed]] [[homotopy type]] $X$ is an interpolation of the point inclusion $* \to X$ by a sequence of homotopy types

\begin{displaymath}
* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} 
  \simeq X
\end{displaymath}
that are obtained from right to left by \emph{removing [[homotopy groups]] from below}, hence such that

\begin{itemize}%
\item each $X^{(n)}$ is $(n-1)$-[[n-simply connected space|connected]]


\item and each [[morphism]] $X^{(n+1)} \to X^{(n)}$ induces an [[isomorphism]] on all [[homotopy groups]] in degree $k \geq (n+1)$ (and the inclusion $1 \to \pi_n(X^{(n)})$ in degree $n$ as well as the identity $1 = 1$ in degree $k \lt n$).



\end{itemize}
The notion of Whitehead tower is [[duality|dual]] to the notion of \emph{[[Postnikov tower]]}, which instead is a factorization of the terminal morphism $X \to *$ into a tower, where homotopy groups are \emph{added} from right to left.

In fact, the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X$ to be the \emph{[[homotopy fiber]]} of the corresponding map into the $(n+1)$st stage of the [[Postnikov tower]].

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

The construction of Whitehead towers is traditionally done for [[topological spaces]] regarded up to [[weak homotopy equivalence]], hence as objects of the [[(∞,1)-category]] [[Top]]. The discussion directly generalizes to any [[(∞,1)-topos]].

The \textbf{Whitehead tower} of a [[homotopy type]] $X$ is a sequence of [[homotopy types]]

\begin{displaymath}
* \to \cdots \to X^{(2)} \to X^{(1)} \to X^{(0)} \simeq X
\end{displaymath}
where [[generalized the|the]] space $X^{(n)}$ is [[generalized the|the]] [[homotopy fiber]] of [[generalized the|the]] map $X \to X_{(n+1)}$ into the item $X_{(n+1)}$ in the [[Postnikov tower]] of $X$.

Here each [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    X^{(n)} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\to& X_{(n+1)}
  }
\end{displaymath}
in the [[(∞,1)-category]] [[Top]] may be computed (as described at [[homotopy pullback]]) as an ordinary [[pullback]] in the 1-[[category]] [[Top]] of a fibrantly replaced diagram, for instance with the point $*$ replaced by the path fibration $P X_{(n+1)} \simeq *$, which is a [[Hurewicz fibration]] $P X_{(n+1)} \to X_{(n+1)}$. In this case also the ordinary [[pullback]] $X^{(n)}\to X$

\begin{displaymath}
\itexarray{
    X^{(n)} &\to& P X_{(n+1)}
    \\
    \downarrow && \downarrow
    \\
    X &\to& X_{(n+1)}
  }
\end{displaymath}
is a fibration, and this is often taken as part of the definition of the Whitehead tower.

From this perspective the \emph{Whitehead tower} of a [[pointed space]] $(X,x)$ is a sequence of fibrations

\begin{displaymath}
\ldots \to X\langle n\rangle \to \ldots \to X\langle 1 \rangle \to X\langle 0 \rangle \to X
\end{displaymath}
where each $X\langle n\rangle \to X\langle n-1 \rangle$ induces isomorphisms on [[homotopy group]]s $\pi_i$ for $i\gt n$ and such that $X\langle n\rangle$ is $n$-[[n-connected|connected]] (has trivial [[homotopy group]]s $\pi_i$ for $i \leq n$). The homotopy long exact sequence then shows that the fiber of $X\langle n\rangle \to X\langle n-1 \rangle$ is a $K(\pi_{n-1}(X,x),n-1)$ [[Eilenberg-Mac Lane space]]. One has a model for $K(\pi_{n-1}(X,x),n-1)$ which is an abelian topological group; this has a remarkable consequence when $(X,x)=(G,e)$ is a [[topological group]]. Indeed, in this case one sees inductively that $G\langle n\rangle$ has a model which is a topological group, which is an abelian group extension:

\begin{displaymath}
1\to K(\pi_{n-1}(X,x),n-1) \to G\langle n\rangle \to G\langle n-1 \rangle \to 1
\end{displaymath}
For instance, the [[string group]] can be realized as a topological group as a $K(\mathbb{Z},2)$-extension of the [[spin group]].

For $n=0$ we require that $X\langle 0 \rangle \hookrightarrow X$ is the inclusion of the path-component of $x$. Really this is defined up to [[homotopy (as an operation)|homotopy]], but we have a canonical model. If $X$ is locally connected and semilocally path-connected, then $X\langle 1\rangle$ can be chosen as the [[universal covering space]].

In traditional models this construction is highly non-[[functor]]ial, except for nice spaces in low dimensions as remarked above.

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

\hypertarget{whiteheads_construction}{}\subsubsection*{{Whitehead's construction}}\label{whiteheads_construction}

\hyperlink{Whitehead}{Whitehead 1952} answered the question, posed by [[Witold Hurewicz]], of the existence of what we would now call $n$-connected 'covers' of a given space $X$, taking this to mean a fibration $X\langle n\rangle \to X$ with $X\langle n\rangle$ $n$-connected and otherwise inducing isomorphisms on homotopy groups.

The construction proceeds as follows (using modern terminology). Given a pointed space $(X,x)$,

\begin{itemize}%
\item Choose a representative for the [[Postnikov section]] $X_n$ such that $X \hookrightarrow X_n$ is a closed subspace (I would be tempted to make it a closed cofibration, but I don't know any reason for this to be necessary -DMR).


\item Form the $\infty$-connected cover of $X_n$, i.e. the [[path fibration]] $P X_n$. This is a [[Hurewicz fibration]].


\item Pull this back to $X$, to get $p\colon X\langle n\rangle \to X$, which is still a fibration. The induced maps on long exact sequences in homotopy can be compared, and show that $p$ has the desired properties.



\end{itemize}
This gives us a single $n$-connected cover, but by considering the [[Postnikov tower]]

\begin{displaymath}
X \to (\ldots \to X_n \to X_{n-1} \to \ldots \to X_1 \to X_0)
\end{displaymath}
of $X$, where each map $X \to X_n$ is the inclusion of a closed subspace, it is simple to see there are induced maps $X\langle n\rangle \to X\langle n-1\rangle$ over $X$ for all $n$.

One way of obtaining a Postnikov section as above is to choose representatives $\phi_g\colon S^{n+1} \to X$ of generators $g$ of $\pi_{n+1}(X,x)$ and attaching cells: $X(1)\coloneqq B^{n+2} \cup_{\{\phi_g\}} X$. We then choose representatives for the generators of $\pi_{n+2}(X(1),x)$ and attach cells and so on. The colimit $\lim_{\to n} X(n)$ is then a Postnikov section with the properties we require.

Understandably, this process is unbelievably non-canonical, and so we are generally reduced to existence theorems using this method -- unless there is a functorial way to construct Postnikov sections. Strictly speaking we can only say \emph{an} $n$-connected cover (except in special cases, like when $n=1$ and $X$ is a [[well-connected space]]).

\hypertarget{functorial_constructions}{}\subsubsection*{{Functorial constructions}}\label{functorial_constructions}

The $n$th stage of the Whitehead tower of $X$ is the [[homotopy fiber]] of the map from $X$ to the $n$th (or so) stage of its [[Postnikov tower]], so one can use a functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the [[path object|path space]] of the target).

The $n$th stage of the Whitehead tower of $X$ is also the cofibrant replacement for $X$ in the right [[Bousfield localization]] of [[Top]] with respect to the object $S^n$ (or so). Since [[Top]] is right proper and cellular this localization exists by the result of chapter 5 of Hirschhorn's book on [[localization]]s of [[model category|model categories]].

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

\hypertarget{OfTheOrthogonalGroup}{}\subsubsection*{{Whitehead tower of the orthogonal group}}\label{OfTheOrthogonalGroup}

The Whitehead tower of the [[classifying space]]/[[delooping]] of the [[orthogonal group]] $O(n)$ starts out as

\begin{displaymath}
\itexarray{
    & \mathbf{\text{Whitehead tower}}
    \\
    &\vdots
    \\
    & B Fivebrane &\to& \cdots &\to& *
    \\
    & \downarrow && && \downarrow
    \\
    \mathbf{\text{second frac Pontr. class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}&  B^8 \mathbb{Z}
    &\to& * 
    \\
    & \downarrow && && \downarrow && \downarrow
    \\
    \mathbf{\text{first frac Pontr. class}} & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * 
    \\
    & \downarrow && && \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\text{second SW class}} & B S O 
    &\to&
    \cdots
    &\to&
    &\to&
    & \stackrel{w_2}{\to} &
    \mathbf{B}^2 \mathbb{Z}_2
    &\to&
    *
    \\
    & \downarrow && && \downarrow && \downarrow &&  \downarrow && \downarrow
    \\
    \mathbf{\text{first SW class}} & B O 
    &\to&
    \cdots
    &\to&
    \tau_{\leq 8 } B O
    &\to&
    \tau_{\leq 4 } B O
    &\to&
    \tau_{\leq 2 } B O
    &\stackrel{w_1}{\to}&
    \tau_{\leq 1 } B O
    \simeq
    B \mathbb{Z}_2
    &
    \mathbf{\text{Postnikov tower}}
  }
\end{displaymath}
where each square and each composite rectangle is a [[homotopy pullback]] square (all controled by the [[pasting law]]),

where the stages are the deloopings of

\ldots{} $\to$ [[fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]],

where lifts through the stages correspond to

\begin{itemize}%
\item [[orthogonal structure]]


\item [[orientation]]


\item [[spin structure]]


\item [[string structure]]


\item [[fivebrane structure]]



\end{itemize}
and where the [[obstruction]] classes are the [[universal characteristic classes]]

\begin{itemize}%
\item [[first Stiefel-Whitney class]] $w_1$


\item [[second Stiefel-Whitney class]] $w_2$


\item [[Pontryagin class|first fractional Pontryagin class]] $\tfrac{1}{2}p_1$


\item [[Pontryagin class|second fractional Pontryagin class]] $\tfrac{1}{6}p_2$



\end{itemize}
and where every possible square in the above is a [[homotopy pullback]] square (using the [[pasting law]]).

For instance $w_2$ can be identified as such by representing $B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}$ by a [[Kan fibration]] (see at [[Postnikov tower]]) between [[Kan complexes]] so that then the [[homotopy pullback]] (as discussed there) is given by an ordinary pullback. Since $sSet$ is a [[simplicial model category]], $sSet(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $B O \to \tau_{\leq 2} B O$ to an isomorphism on connected components. This identifies $B SO \to B^2 \mathbb{Z}$ as being an [[isomorphism]] on the second [[homotopy group]]. Therefore, by the [[Hurewicz theorem]], it is also an isomorphism on the [[cohomology group]] $H^2(-,\mathbb{Z}_2)$. Analogously for the other characteristic maps.

In summary, more concisely, the tower is

\begin{displaymath}
\itexarray{
    \vdots
    \\
    \downarrow
    \\
    B Fivebrane
    \\
    \downarrow
    \\
    B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z}
    \\
    \downarrow
    \\
    B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z}
    \\
    \downarrow
    \\
    B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2
    \\
    \downarrow
    \\
    B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    B GL
  }
  \,,
\end{displaymath}
where each ``hook'' is a [[fiber sequence]].

Via the [[J-homomorphism]] this corresponds to the [[stable homotopy groups of spheres]]:

[[!include image of J -- table]]

\hypertarget{whitehead_tower_in_general_toposes}{}\subsection*{{Whitehead tower in general $(\infty,1)$-toposes}}\label{whitehead_tower_in_general_toposes}

While a notion of [[Postnikov tower in an (∞,1)-category]] depends on the \emph{categorical} [[homotopy groups in an (∞,1)-topos|homotopy groups in an (∞,1)-category]], the notion of Whitehead tower makes good sense with respect to the \emph{geometric} homotopy groups.

A good notion of geometric [[homotopy groups in an (∞,1)-topos]] exist in a [[locally contractible (∞,1)-topos]]. The notion of Whitehead tower in this context is discussed at

\begin{itemize}%
\item [[Whitehead tower in an (∞,1)-topos]].

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

\begin{itemize}%
\item Applying the [[Hurewicz theorem]] stagewise to a [[Whitehead tower]] yields an method for computing the [[homotopy groups]] of the original [[space]]. This process, or rather the refinement thereof for Whitehead towers generalized to [[Adams resolutions]], is formalized by the \emph{[[Adams spectral sequence]]}, see there for more.


\item \emph{\href{https://ncatlab.org/nlab/show/permutation#WhietheadTowerAndSupersymmetry}{symmetric group -- Whitehead tower}}



\end{itemize}
[[!include Lurie spectral sequences -- table]]

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

The original reference is

\begin{itemize}%
\item [[George Whitehead]] \emph{Fiber Spaces and the Eilenberg Homology Groups}, PNAS \textbf{38}, No. 5 (1952)

\end{itemize}
A textbook account is around example 4.20 in

\begin{itemize}%
\item [[Allen Hatcher]], \emph{Algebraic Topology} (\href{http://www.math.cornell.edu/~hatcher/AT/AT.pdf}{pdf})

\end{itemize}
A more detailed useful discussion happens to be in section 2.A, starting on p. 11 of

\begin{itemize}%
\item [[Linus Kramer]], \emph{Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface} Memoirs of the American Mathematical Society number 752 (\href{http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false}{web}) also (\href{http://arxiv.org/abs/math/0109133}{arXiv})

\end{itemize}
[[!redirects Whitehead tower]] [[!redirects Whitehead towers]]



\end{document}