\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*{iterated loop space}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{relation_to_configuration_spaces_of_points}{Relation to configuration spaces of points}\dotfill \pageref*{relation_to_configuration_spaces_of_points} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_configuration_spaces_of_points_2}{Relation to configuration spaces of points}\dotfill \pageref*{relation_to_configuration_spaces_of_points_2} \linebreak
\noindent\hyperlink{rational_cohomology}{Rational cohomology}\dotfill \pageref*{rational_cohomology} \linebreak


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

A multiple [[loop space]].

A grouplike [[E-k algebra]] in [[Top]].

An [[iterated loop space object]] in [[Top]].

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

\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

\begin{prop}
\label{RationalCohomologyOfIteratedLoopSpaceOf2kSphere}\hypertarget{RationalCohomologyOfIteratedLoopSpaceOf2kSphere}{}
\textbf{([[rational cohomology]] of [[iterated loop space]] of the [[n-sphere|2k-sphere]])}

Let

\begin{displaymath}
1 \leq D \lt n = 2k \in \mathbb{N}
\end{displaymath}
(hence two [[positive number|positive]] [[natural numbers]], one of them required to be [[even number|even]] and the other required to be smaller than the first) and consider the [[iterated loop space|D-fold loop space]] $\Omega^D S^n$ of the [[n-sphere]].

Its [[rational cohomology|rational]] [[cohomology ring]] is the [[free construction|free]] [[graded-commutative algebra]] over $\mathbb{Q}$ on one [[generators and relations|generator]] $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:

\begin{displaymath}
H^\bullet
  \big(
    \Omega^D S^n
    ,
    \mathbb{Q}
  \big)
  \;\simeq\;
  \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big] 
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{KallelSjerve99}{Kallel-Sjerve 99, Prop. 4.10})

$\backslash$linebreak

\hypertarget{relation_to_configuration_spaces_of_points}{}\subsubsection*{{Relation to configuration spaces of points}}\label{relation_to_configuration_spaces_of_points}

\begin{prop}
\label{ScanningMapEquivalenceOverCartesianSpace}\hypertarget{ScanningMapEquivalenceOverCartesianSpace}{}
\textbf{([[iterated loop spaces]] equivalent to [[configuration spaces of points]])}

For

\begin{enumerate}%
\item $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],


\item $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that $Y / \partial Y$ is [[connected topological space|connected]],



\end{enumerate}
the electric field map/[[scanning map]] constitutes a [[homotopy equivalence]]

\begin{displaymath}
Conf\left( 
    \mathbb{R}^d, Y
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d \Sigma^d (Y/\partial Y)
\end{displaymath}
between

\begin{enumerate}%
\item the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. \ref{ConfigurationSpacesOfnPoints})


\item the [[iterated loop space|d-fold loop space]] of the $d$-fold [[reduced suspension]] of the [[quotient space]] $Y / \partial Y$ (regarded as a [[pointed topological space]] with basepoint $[\partial Y]$).



\end{enumerate}
In particular when $Y = \mathbb{D}^k$ is the [[closed ball]] of [[dimension]] $k \geq 1$ this gives a [[homotopy equivalence]]

\begin{displaymath}
Conf\left( 
    \mathbb{R}^d, \mathbb{D}^k
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d S^{ d + k }
\end{displaymath}
with the [[iterated loop space|d-fold loop space]] of the [[n-sphere|(d+k)-sphere]].

\end{prop}
(\hyperlink{May72}{May 72, Theorem 2.7}, \hyperlink{Segal73}{Segal 73, Theorem 3}, see \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 13})

\begin{prop}
\label{StableSplittingOfMappingSpacesOutOfEuclideanSpace}\hypertarget{StableSplittingOfMappingSpacesOutOfEuclideanSpace}{}
\textbf{([[stable splitting of mapping spaces]] out of [[Euclidean space]]/[[n-spheres]])}

For

\begin{enumerate}%
\item $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],


\item $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that $Y / \partial Y$ is [[connected topological space|connected]],



\end{enumerate}
there is a [[stable weak homotopy equivalence]]

\begin{displaymath}
\Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
\end{displaymath}
between

\begin{enumerate}%
\item the [[suspension spectrum]] of the [[configuration space of points|configuration space]] of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})


\item the [[direct sum]] (hence: [[wedge sum]]) of [[suspension spectra]] of the [[configuration space of points|configuration spaces]] of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).



\end{enumerate}
Combined with the [[stabilization]] of the electric field map/[[scanning map]] [[homotopy equivalence]] from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a [[stable weak homotopy equivalence]]

\begin{displaymath}
Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  =
  \Omega^d \Sigma^d (Y/\partial Y)
  \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow}
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
\end{displaymath}
between the latter direct sum and the [[suspension spectrum]] of the [[mapping space]] of pointed [[continuous functions]] from the [[n-sphere|d-sphere]] to the $d$-fold [[reduced suspension]] of $Y / \partial Y$.

\end{prop}
(\hyperlink{Snaith74}{Snaith 74, theorem 1.1}, \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 2})

In fact by \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 5} this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

\begin{displaymath}
Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include k-monoidal table]]

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

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

\begin{itemize}%
\item [[Peter May]], \emph{Infinite loop space theory}, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (\href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538891}{Euclid})

\emph{Infinite loop space theory revisited} (\href{http://www.math.uchicago.edu/~may/PAPERS/28.pdf}{pdf})


\item [[John Adams]], \emph{Infinite loop spaces}, Herrmann Weyl lectures at IAS, Princeton University Press (1978)


\item [[Peter May]], \emph{The uniqueness of infinite loop space machines}, Topology, vol 17, pp. 205-224 (1978) (\href{http://www.math.uchicago.edu/~may/PAPERS/22.pdf}{pdf})


\item [[Jacob Lurie]], Section 5.1.3 of \emph{[[Higher Algebra]]}



\end{itemize}
\hypertarget{relation_to_configuration_spaces_of_points_2}{}\subsubsection*{{Relation to configuration spaces of points}}\label{relation_to_configuration_spaces_of_points_2}

In relation to [[configuration spaces of points]]:

\begin{itemize}%
\item [[Peter May]], \emph{The geometry of iterated loop spaces}, Springer 1972 (\href{https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf}{pdf})


\item [[Graeme Segal]], \emph{Configuration-spaces and iterated loop-spaces}, Invent. Math. \textbf{21} (1973), 213--221. MR 0331377 (\href{http://dodo.pdmi.ras.ru/~topology/books/segal.pdf}{pdf})


\item [[Victor Snaith]], \emph{A stable decomposition of $\Omega^n S^n X$}, Journal of the London Mathematical Society 7 (1974), 577 - 583 (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf}{pdf})


\item [[Dusa McDuff]], \emph{Configuration spaces of positive and negative particles}, Topology Volume 14, Issue 1, March 1975, Pages 91-107 ()


\item [[Carl-Friedrich Bödigheimer]], \emph{Stable splittings of mapping spaces}, Algebraic topology. Springer 1987. 174-187 (\href{http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf}{pdf}, [[BoedigheimerStableSplittings87.pdf:file]])



\end{itemize}
\hypertarget{rational_cohomology}{}\subsubsection*{{Rational cohomology}}\label{rational_cohomology}

On [[ordinary cohomology]] of iterated loop spaces in relation to [[configuration spaces of points]] (see also at \emph{[[graph complex]]}):

\begin{itemize}%
\item [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, \emph{On the homology of configuration spaces}, Topology Vol. 28 No. 1, p. 111-123 1989 (\href{https://core.ac.uk/download/pdf/82124359.pdf}{pdf})

\end{itemize}
On the [[rational cohomology]]:

\begin{itemize}%
\item [[Sadok Kallel]], [[Denis Sjerve]], \emph{On Brace Products and the Structur eof Fibrations with Section}, 1999 (\href{https://www.math.ubc.ca/~sjer/brace.pdf}{pdf}, [[KallelSjerv99.pdf:file]])

\end{itemize}
[[!redirects iterated loop spaces]]

[[!redirects iterated based loop space]] [[!redirects iterated based loop spaces]]

[[!redirects n-fold loop space]] [[!redirects n-fold loop spaces]]



\end{document}