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\begin{document}

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\section*{configuration space of points}

\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{ordered_unlabeled_points}{Ordered unlabeled points}\dotfill \pageref*{ordered_unlabeled_points} \linebreak
\noindent\hyperlink{unordered_unlabeled_points}{Unordered unlabeled points}\dotfill \pageref*{unordered_unlabeled_points} \linebreak
\noindent\hyperlink{UnorderedLabeledPoints}{Unordered labeled points}\dotfill \pageref*{UnorderedLabeledPoints} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations}{Ordered unlabeled configurations from unordered labeled configurations}\dotfill \pageref*{OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations} \linebreak
\noindent\hyperlink{cohomotopy_charge_map}{Cohomotopy charge map}\dotfill \pageref*{cohomotopy_charge_map} \linebreak
\noindent\hyperlink{LoopSpacesOfSuspensions}{Relation to iterated loop spaces of iterated suspensions}\dotfill \pageref*{LoopSpacesOfSuspensions} \linebreak
\noindent\hyperlink{relation_to_classifying_space_of_the_symmetric_group}{Relation to classifying space of the symmetric group}\dotfill \pageref*{relation_to_classifying_space_of_the_symmetric_group} \linebreak
\noindent\hyperlink{relation_to_james_construction}{Relation to James construction}\dotfill \pageref*{relation_to_james_construction} \linebreak
\noindent\hyperlink{RelationToTwistedCohomotopy}{In twisted Cohomotopy}\dotfill \pageref*{RelationToTwistedCohomotopy} \linebreak
\noindent\hyperlink{action_by_little_disk_operad_and_by_goodwillie_derivatives}{Action by little $n$-disk operad and by Goodwillie derivatives}\dotfill \pageref*{action_by_little_disk_operad_and_by_goodwillie_derivatives} \linebreak
\noindent\hyperlink{HomologyAndStabilization}{Homology and stabilization in homology}\dotfill \pageref*{HomologyAndStabilization} \linebreak
\noindent\hyperlink{RationalHomotopyType}{Rational homotopy type}\dotfill \pageref*{RationalHomotopyType} \linebreak
\noindent\hyperlink{Cohomology}{Rational cohomology}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{RationalHomotopyGroups}{Rational homotopy and Whitehead products}\dotfill \pageref*{RationalHomotopyGroups} \linebreak
\noindent\hyperlink{OccurrencesAndApplications}{Occurrences and Applications}\dotfill \pageref*{OccurrencesAndApplications} \linebreak
\noindent\hyperlink{compactification}{Compactification}\dotfill \pageref*{compactification} \linebreak
\noindent\hyperlink{stable_splitting_of_mapping_spaces}{Stable splitting of mapping spaces}\dotfill \pageref*{stable_splitting_of_mapping_spaces} \linebreak
\noindent\hyperlink{correlators_as_differential_forms_on_configuration_spaces}{Correlators as differential forms on configuration spaces}\dotfill \pageref*{correlators_as_differential_forms_on_configuration_spaces} \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{cohomotopy_charge_map_2}{Cohomotopy charge map}\dotfill \pageref*{cohomotopy_charge_map_2} \linebreak
\noindent\hyperlink{stable_splitting_of_mapping_spaces_2}{Stable splitting of mapping spaces}\dotfill \pageref*{stable_splitting_of_mapping_spaces_2} \linebreak
\noindent\hyperlink{ReferencesInGoodwillieCalculus}{In Goodwillie-calculus}\dotfill \pageref*{ReferencesInGoodwillieCalculus} \linebreak
\noindent\hyperlink{ReferencesCompactification}{Compactification}\dotfill \pageref*{ReferencesCompactification} \linebreak
\noindent\hyperlink{ReferencesCohomology}{Homology and cohomology}\dotfill \pageref*{ReferencesCohomology} \linebreak
\noindent\hyperlink{homotopy}{Homotopy}\dotfill \pageref*{homotopy} \linebreak
\noindent\hyperlink{rational_homotopy_type_2}{Rational homotopy type}\dotfill \pageref*{rational_homotopy_type_2} \linebreak
\noindent\hyperlink{cohomology_modeled_by_graph_complexes}{Cohomology modeled by graph complexes}\dotfill \pageref*{cohomology_modeled_by_graph_complexes} \linebreak
\noindent\hyperlink{ReferencesLoopSpacesOfConfigurationSpaces}{Loop spaces of configuration spaces of points}\dotfill \pageref*{ReferencesLoopSpacesOfConfigurationSpaces} \linebreak
\noindent\hyperlink{AsModuliOfD0D4BraneBoundStates}{As moduli of Dp-D(p+4)-brane bound states:}\dotfill \pageref*{AsModuliOfD0D4BraneBoundStates} \linebreak


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

In [[mathematics]], the term ``configuration space'' of a [[topological space]] $X$ typically refers by default to the topological space of pairwise distinct [[element|points]] in $X$, also called \emph{[[Fadell's configuration space]]}, for emphasis.

In principle many other kinds of configurations and the spaces these form may be referred to by ``configuration space'', notably in [[physics]] the usage is in a broader sense, see at \emph{[[configuration space (physics)]]}.

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

Several variants of configuration spaces of points are of interest. They differ in whether

\begin{enumerate}%
\item points are linearly ordered or not;


\item points are labeled in some labelling space;


\item points vanish on some subspace or if their labels are in some subspace.



\end{enumerate}
Here are some of these variant definitions:

\hypertarget{ordered_unlabeled_points}{}\subsubsection*{{Ordered unlabeled points}}\label{ordered_unlabeled_points}

\begin{defn}
\label{OrderedUnlabeledConfigurations}\hypertarget{OrderedUnlabeledConfigurations}{}
\textbf{(ordered unlabled configurations of a fixed number of points)}

Let $X$ be a [[closed manifold|closed]] [[smooth manifold]]. For $n \in \mathbb{N}$ write

\begin{displaymath}
\underset{ {}^{\{1,\cdots, n\}} }{ Conf }
  \big( 
    X
  \big)
  \;\coloneqq\;
  \big(
    X
  \big)^n
  \setminus
  \mathbf{\Delta}^n_X
\end{displaymath}
for the [[complement]] of the [[fat diagonal]] inside the $n$-fold [[Cartesian product]] of $X$ with itself.

This is the space of ordered but otherwise unlabeled configurations of $n$ points\_ in $X$.

\end{defn}
$\backslash$linebreak

\hypertarget{unordered_unlabeled_points}{}\subsubsection*{{Unordered unlabeled points}}\label{unordered_unlabeled_points}

\begin{defn}
\label{UnorderedUnlabeledConfigurations}\hypertarget{UnorderedUnlabeledConfigurations}{}
\textbf{(unordered unlabled configurations of a fixed number of points)}

Let $X$ be a [[closed manifold|closed]] [[smooth manifold]], For $n \in \mathbb{N}$ write

\begin{equation}
\begin{aligned}
    Conf_n
    \big(
      X
    \big)
    &
    \coloneqq
    \;
    \Big(
      \underset{{}^{1,\cdots,n}}{Conf} 
      \big(
        X
      \big)
    \big) / Sym(n)
    \\
    &
    =\;
    \Big(
      \big(
        X
      \big)^n
      \setminus
      \mathbf{\Delta}^n_X
    \Big) / Sym(n)
  \end{aligned}
\label{UnorderedUnlabelednPointConfigurationSpaces}\end{equation}
for the [[quotient space]] of the ordered configuration space (Def. \ref{OrderedUnlabeledConfigurations}) by the evident [[action]] of the [[symmetric group]] $Sym(n)$ via [[permutation]] of the [[ordering]] of the points.

This is the space of unordered and unlabeled configurations of $n$ points\_ in $X$.

We write

\begin{equation}
Conf(X)
  \;\coloneqq\;
  \underset{n \in \mathbb{N}}{\sqcup}
  Conf_n\big( X\big)
\label{UnorderedUnlabeledConfigurationSpace}\end{equation}
for the unordered unlabeled configuration space of any [[finite number]] of points, being the [[disjoint union]] of these spaces \eqref{UnorderedUnlabelednPointConfigurationSpaces} over all [[natural numbers]] $n$.

\end{defn}
\begin{remark}
\label{MonoidStructureAndItsGroupCompletion}\hypertarget{MonoidStructureAndItsGroupCompletion}{}
\textbf{([[monoid]]-[[mathematical structure|structure]] on [[configuration space of points]])}

For $X = \mathbb{R}^D$ a [[Euclidean spaces]] the configuration space of points $Conf\big( \mathbb{R}^D \big)$ \eqref{UnorderedUnlabeledConfigurationSpace} carriesthe [[mathematical structure|structure]] of a [[topological monoid]] with product operation being the [[disjoint union]] of point configurations, after a suitable shrinking to put them next to each other (\hyperlink{Segal73}{Segal 73, p. 1-2}).

For emphasis, we write $B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D)$ for the [[delooping]] (``[[classifying space]]'') with respect to this [[topological monoid]]-[[structure]]. The corresponding [[based loop space]] is then the [[group completion]] of the configuration space, with respect to disjoint union of points:

\begin{equation}
Conf
  \big( 
    \mathbb{R}^D
  \big)
  \overset{\color{blue}\text{group completion}}{\longrightarrow}
  \Omega B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D)
  \,.
\label{GroupCompletionOfConfigurationSpaceMonoid}\end{equation}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The configuration space of unordered unlabeled configurations of $n$ points (Def. \ref{OrderedUnlabeledConfigurations}) is naturally a [[topological subspace]] of the [[space of finite subsets]] of [[cardinality]] $\leq n$

\begin{equation}
Conf_n(X)
  \hookrightarrow
  \exp^n(X)
\label{InjectionOfUnorderedConfigurationSpaceOfPoints}\end{equation}
\end{remark}
\begin{prop}
\label{UnorderedConfigurationSpaceInSpaceOfFiniteSubsets}\hypertarget{UnorderedConfigurationSpaceInSpaceOfFiniteSubsets}{}
Let $X$ be an [[inhabited set|non-empty]] [[regular topological space]] and $n \geq 2 \in \mathbb{N}$.

Then the injection \eqref{InjectionOfUnorderedConfigurationSpaceOfPoints}

\begin{equation}
Conf_n(X)
  \hookrightarrow
  \exp^n(X)/\exp^{n-1}(X)
\label{UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets}\end{equation}
of the [[unordered configuration space of n points]] of $X$ (Def. \ref{OrderedUnlabeledConfigurations}) into the [[quotient space]] of the [[space of finite subsets]] of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an [[open subspace]]-inclusion.

Moreover, if $X$ is [[compact topological space|compact]], then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion \eqref{UnorderedConfigurationSpaceOpenSubsetInQuotientOfSpacesOfFiniteSubsets} exhibits the [[one-point compactification]] $\big(  Conf_n(X) \big)^{+}$ of the configuration space:

\begin{displaymath}
\big( 
    Conf_n(X)
  \big)^{+}
  \;\simeq\;
  \exp^n(X)/\exp^{n-1}(X)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Handel00}{Handel 00, Prop. 2.23}, see also \hyperlink{FelixTanre10}{Félix-Tanré 10})

\hypertarget{UnorderedLabeledPoints}{}\subsubsection*{{Unordered labeled points}}\label{UnorderedLabeledPoints}

\begin{defn}
\label{UnorderedLabeledFixedn}\hypertarget{UnorderedLabeledFixedn}{}
For $X$ a [[smooth manifold]] and $k \in \mathbb{N}$, the space of \emph{unordered configurations of points in $X$ with labels in $S^k$} is

\begin{equation}
Conf_n\big(X, S^k \big)
  \;\coloneqq\;
  Conf_n\big(X\big) \underset{Sym(n)}{\times}
  \big(
    S^k 
  \big)^n
\label{UnorderedLabeledFixednCOnfigurationSpace}\end{equation}
\end{defn}
For $k \in \mathbb{N}$, consider the [[n-sphere|k-sphere]] as a [[pointed topological space]], with the base point regarded as the ``vanishing label''.

\begin{defn}
\label{UnorderedLabeledConfigurationsVanishingWithVanishingLabel}\hypertarget{UnorderedLabeledConfigurationsVanishingWithVanishingLabel}{}
\textbf{(unordered labeled configurations vanishing with vanishing label)}

For $X$ a [[smooth manifold]] and $k \in \mathbb{N}$, the space of \emph{unordered configurations of points in $X$ with labels in $S^k$ and vanishing at vanishing label value} is the [[quotient space]]

\begin{equation}
Conf
  \big( 
    X,
    S^k
  \big)
  \;\coloneqq\;
  \Big(
    \underset{n \in \mathbb{N}}{\sqcup}
    Conf_n\big(X,S^k \big)
  \Big)/\sim
\label{UnorderedLabeledCOnfigurationSpace}\end{equation}
of the [[disjoint union]] of all unordred labeled $n$-point configuration spaces \eqref{UnorderedLabeledFixednCOnfigurationSpace} by the [[equivalence relation]] which regards points with vanishing label as absent.

\end{defn}
\begin{defn}
\label{ConfigurationSpacesOfnPoints}\hypertarget{ConfigurationSpacesOfnPoints}{}
\textbf{(unordered labeled configurations of a fixed number of points)}

Let $X$ be a [[manifold]], possibly with [[manifold with boundary|boundary]]. For $n \in \mathbb{N}$, the \emph{\textbf{configuration space of $n$ unordered points} in $X$ disappearing at the boundary} is the [[topological space]]

\begin{displaymath}
\mathrm{Conf}_{n}(X)
    \;\coloneqq\;
    \Big(
      \big(
        X^n \setminus \mathbf{\Delta}_X^n
      \big)
      / \partial(X^n)
    \Big)
    /\Sigma(n)
    \,,
\end{displaymath}
where $\mathbf{\Delta}_X^n : = \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ is the [[fat diagonal]] in $X^n$ and where $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X$ inside $X^n$.

More generally, let $Y$ be another [[manifold]], possibly with [[manifold with boundary|boundary]]. For $n \in \mathbb{N}$, the \emph{\textbf{configuration space of $n$ points} in $X \times Y$ vanishing at the boundary and distinct as points in $X$} is the [[topological space]]

\begin{displaymath}
\mathrm{Conf}_{n}(X,Y)
    \;\coloneqq\;
    \Big(
    \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^n
    \big)
    /\Sigma(n)
    \Big)
    / \partial(X^n \times Y^n)
\end{displaymath}
where now $\Sigma(n)$ denotes the evident [[action]] of the [[symmetric group]] by [[permutation]] of factors of $X \times Y$ inside $X^n \times Y^n \simeq (X \times Y)^n$.

This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:

\begin{displaymath}
\mathrm{Conf}_n(X)
    \;=\;
    \mathrm{Conf}_n(X,\ast)
    \,.
\end{displaymath}
Finally the \emph{\textbf{configuration space of an arbitrary number of points} in $X \times Y$ vanishing at the boundary and distinct already as points of $X$} is the [[quotient topological space]] of the [[disjoint union space]]

\begin{displaymath}
Conf\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{n \in \mathbb{n}}{\sqcup}
      \big(
      (
        X^n \setminus \mathbf{\Delta}_X^n
      )
      \times
      Y^k
    \big)
    /\Sigma(n)
  \right)/\sim
\end{displaymath}
by the [[equivalence relation]] $\sim$ given by

\begin{displaymath}
\big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n)
  \big)
  \;\sim\;
  \big(
    (x_1, y_1), \cdots, (x_{n-1}, y_{n-1})
  \big)
  \;\;\;\; \Leftrightarrow
  \;\;\;\; (x_n, y_n) \in \partial (X \times Y)
  \,.
\end{displaymath}
This is naturally a [[filtered topological space]] with filter stages

\begin{displaymath}
Conf_{\leq n}\left( X, Y\right)
  \;\coloneqq\;
  \left(
  \underset{k \in \{1, \cdots, n\}}{\sqcup}
      \big(
      (
        X^k \setminus \mathbf{\Delta}_X^k
      )
      \times
      Y^k
    \big)
    /\Sigma(k)
  \right)/\sim
  \,.
\end{displaymath}
The corresponding [[quotient topological spaces]] of the filter stages reproduces the above configuration spaces of a fixed number of points:

\begin{displaymath}
Conf_n(X,Y)
  \;\simeq\;
  Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{(comparison to notation in the literature)}

The above Def. \ref{ConfigurationSpacesOfnPoints} is less general but possibly more suggestive than what is considered for instance in \hyperlink{Boedigheimer87}{Bödigheimer 87}. Concretely, we have the following translations of notation:

\begin{displaymath}
\itexarray{
      \text{ here: }
        && 
      \itexarray{ \text{ Segal 73,} \\ \text{ Snaith 74}: }
        &&
      \text{ Bödigheimer 87: }
      \\
      \\
      Conf(\mathbb{R}^d,Y) 
        &=& 
      C_d( Y/\partial Y )
        &=&
      C( \mathbb{R}^d, \emptyset; Y )  
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d \right)
      & = &
      F_n C_d( S^0 ) / F_{n-1} C_d( S^0 )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; S^0  \right)
      \\
      \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right)
      & = &
      F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y )
      & = &
      D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y  \right)
      \\
      \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0  \right) 
      \\
      \mathrm{Conf}_n( X, Y  ) && &=& D_n\left( X, \partial X; Y/\partial Y  \right)
    }
\end{displaymath}
Notice here that when $Y$ happens to have [[empty space|empty]] [[boundary]], $\partial Y = \emptyset$, then the [[pushout]]

\begin{displaymath}
X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast
\end{displaymath}
is $Y$ with a \href{pointed+topological+space#ForgettingAndAdjoiningBasepoints}{disjoint basepoint attached}. Notably for $Y =\ast$ the [[point space]], we have that

\begin{displaymath}
\ast/\partial \ast = S^0
\end{displaymath}
is the [[0-sphere]].

\end{remark}
A slight variation of the definition is sometimes useful:

\begin{defn}
\label{ConfigurationSpaceOfDisks}\hypertarget{ConfigurationSpaceOfDisks}{}
\textbf{(configuration space of $dim(X)$-disks)}

In the situation of Def. \ref{ConfigurationSpaceOfParticleswithLabels}, with $X$ a [[manifold]] of [[dimension]] $dim(X) \in \mathbb{N}$

\begin{displaymath}
DiskConf(X,A)
  \longrightarrow
  Conf(X,A)
\end{displaymath}
be, on the left, the labeled configuration space of joint [[embedding of smooth manifolds|embeddings]] of [[tuples]]

\begin{displaymath}
\left(
    D^{dim(X)} 
      \overset{ \iota_i }{\hookrightarrow}
    X
  \right)
\end{displaymath}
of $dim(X)$-dimensional disks/[[closed balls]] into $X$, with identifications as in Def. \ref{ConfigurationSpaceOfParticleswithLabels} (in particular the disks centered at the basepoint are quotiented out) and with the comparison map sending each disk to its center.

This map is evidently a [[deformation retraction]] hence in particular a [[homotopy equivalence]].

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

\hypertarget{OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations}{}\subsubsection*{{Ordered unlabeled configurations from unordered labeled configurations}}\label{OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations}

\begin{quote}%
under construction


\end{quote}
(\ldots{})

(\ldots{})

\hypertarget{cohomotopy_charge_map}{}\subsubsection*{{Cohomotopy charge map}}\label{cohomotopy_charge_map}

The \emph{Cohomotopy charge map} is the [[function]] that assigns to a [[configuration space of points|configuration of points]] their total [[charge]] as measured in [[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

This is alternatively known as the ``electric field map'' (\hyperlink{Salvatore01}{Salvatore 01} following \hyperlink{Segal73}{Segal 73, Section 1}, see also \hyperlink{Knudsen18}{Knudsen 18, p. 49}) or the ``scanning map'' (\hyperlink{Kallel98}{Kallel 98}).

For $D \in \mathbb{N}$ the \emph{Cohomotopy charge map} is the [[continuous function]]

\begin{equation}
Conf\big( \mathbb{R}^D \big)
  \overset{cc}{\longrightarrow}
  \mathbf{\pi}^D
  \Big( 
    \big( \mathbb{R}^D \big)^{cpt}
  \Big)
  =
  Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big)
  =
  \Omega^{D} S^D
\label{CohomotopyChargeMapOnEuclideanSpace}\end{equation}
from the [[configuration space of points]] in the [[Euclidean space]] $\mathbb{R}^D$ to the $D$-[[Cohomotopy]] [[cocycle space]] [[vanishing at infinity]] on the [[Euclidean space]], which is equivalently the [[space of maps|space of pointed maps]] from the [[one-point compactification]] $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]]), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit [[charge]] to their total [[charge]] as measured in [[Cohomotopy]]-[[generalized cohomology|cohomology theory]] (\hyperlink{Segal73}{Segal 73, Section 3}).

The construction has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to [[equivariant Cohomotopy]]. The following graphics illustrates the Cohomotopy charge map on [[G-space]] [[tori]] for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-[[equivariant Cohomotopy]]:

\begin{quote}%
graphics grabbed from \href{Cohomotpy+charge+map#SatiSchreiber19}{SS 19}


\end{quote}
\hypertarget{LoopSpacesOfSuspensions}{}\paragraph*{{Relation to iterated loop spaces of iterated suspensions}}\label{LoopSpacesOfSuspensions}

In some situations the [[Cohomotopy charge map]] is a [[weak homotopy equivalence]] and hence exhibits, for all purposes of [[homotopy theory]], the [[Cohomotopy]] [[cocycle space]] of Cohomotopy charges as an equivalent reflection of the [[configuration space of points]]:

\begin{prop}
\label{GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace}\hypertarget{GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace}{}
\textbf{([[group completion]] on [[configuration space of points]] is [[iterated based loop space]])}

The [[Cohomotopy charge map]] \eqref{CohomotopyChargeMapOnEuclideanSpace}

\begin{displaymath}
Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    cc
  }{\longrightarrow}
  \Omega^D S^D
\end{displaymath}
from the full unordered and unlabeled configuration space \eqref{UnorderedUnlabeledConfigurationSpace} of [[Euclidean space]] $\mathbb{R}^D$ to the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]], exhibits the [[group completion]] \eqref{GroupCompletionOfConfigurationSpaceMonoid} of the configuration space monoid

\begin{displaymath}
\Omega B_{{}_{\sqcup}\!} 
  Conf
  \big(
    \mathbb{R}^D
  \big)
  \overset{
    \simeq
  }{\longrightarrow}
  \Omega^D S^D
\end{displaymath}
\end{prop}
(\hyperlink{Segal73}{Segal 73, Theorem 1})

\begin{prop}
\label{CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace}\hypertarget{CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace}{}
\textbf{([[Cohomotopy charge map]] is [[weak homotopy equivalence]] on sphere-labeled [[configuration space of points]])}

For $D, k \in \mathbb{N}$ with $k \geq 1$, the [[Cohomotopy charge map]] \eqref{CohomotopyChargeMapOnEuclideanSpace}

\begin{displaymath}
Conf
  \big(
    \mathbb{R}^D,
    S^k
  \big)
  \underoverset{\simeq}{\;\;cc\;\;}{\longrightarrow}
  \Omega^D S^{D + k}
  \;\simeq\;
  \mathbf{\pi}^{D+ k}\Big( \big( \mathbb{R}^{D}\big)^{cpt} \Big)
\end{displaymath}
is a [[weak homotopy equivalence]]

\begin{itemize}%
\item from the configuration space \eqref{UnorderedLabeledCOnfigurationSpace} of unordered points with labels in $S^k$ and vanishing at the base point of the label space


\item to the $D$-fold [[iterated loop space]] of the [[n-sphere|D+k-sphere]]



\end{itemize}
hence equivalently

\begin{itemize}%
\item to the [[cocycle space]] of [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]] in degree $D + k$ [[vanishing at infinity]] on [[Euclidean space]] of [[dimension]] $D$.

\end{itemize}
\end{prop}
(\hyperlink{Segal73}{Segal 73, Theorem 3})

This statement generalizes to [[equivariant homotopy theory]], with equivariant configurations carrying charge in [[equivariant Cohomotopy]]:

Let $G$ be a [[finite group]] and $V \in RO(G)$ an [[orthogonal group|orthogonal]] $G$-[[linear representation]], with its induced [[pointed topological space|pointed]] [[topological G-spaces]]:

\begin{enumerate}%
\item the corresponding [[representation sphere]] $S^V \in G TopSpaces$,


\item the corresponding [[Euclidean G-space]] $\mathbb{R}^V \in G TopSpaces$.



\end{enumerate}
For $X \in G TopSpaces$ any [[pointed topological space|pointed]] [[topological G-space]], consider

\begin{enumerate}%
\item the equivariant $V$-[[suspension]], given by the [[smash product]] with the $V$-[[representation sphere]]:

$\Sigma^V X \;\coloneqq\; X \wedge S^V \;\in G TopSpaces\;$


\item the equivariant $V$-[[iterated based loop space]], given by the $G$-[[fixed point subspace]] inside the [[space of maps]] out of the [[representation sphere]]:

$\Omega^V X \;\coloneqq\; Maps^{\ast/}\big( S^V, X\big)^G$.



\end{enumerate}
\begin{defn}
\label{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}\hypertarget{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}{}
\textbf{(equivariant unordered labeled configurations vanishing with vanishing label)}

Write

\begin{displaymath}
Conf\big( \mathbb{R}^V , X \big)^G
  \;\hookrightarrow\;
  Conf\big( \mathbb{R}^V , X \big)
\end{displaymath}
for the $G$-[[fixed point subspace]] in the unordered $X$-labelled configuration space of points (Def. \ref{UnorderedLabeledConfigurationsVanishingWithVanishingLabel}), with respect to the [[diagonal action]] on $\mathbb{R}^V \times X$.

\end{defn}
\begin{prop}
\label{EquivariantCohomotopyChargeMapEquivalence}\hypertarget{EquivariantCohomotopyChargeMapEquivalence}{}
\textbf{([[Cohomotopy charge map]]-equivalence for configurations on [[Euclidean G-spaces]])}

Let

\begin{enumerate}%
\item $G$ be a [[finite group]],


\item $V$ an [[orthogonal group|orthogonal]] $G$-[[linear representation]]


\item $X$ a [[topological G-space]]



\end{enumerate}
If $X$ is $G$-connected, in that for all [[subgroups]] $H \subset G$ the $H$-[[fixed point subspace]] $X^H$ is a [[connected topological space]], then the [[Cohomotopy charge map]]

\begin{displaymath}
Conf
  \big(
    \mathbb{R}^V,
    X
  \big)
  \underoverset{\simeq}{\;cc\;}{\longrightarrow}
  \Omega^V \Sigma^V X
  \phantom{AAA}
  \text{if X is G-connected}
\end{displaymath}
from the equivariant un-ordered $X$-labeled configuration space of points (Def. \ref{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}) in the corresponding [[Euclidean G-space]] to the based $V$-loop space of the $V$-suspension of $X$, is a [[weak homotopy equivalence]].

If $X$ is not necessarily $G$-connected, then this still holds for the [[group completion]] of the configuration space, under disjoint union of configurations

\begin{displaymath}
\Omega 
  B_{{}_{\sqcup}\!}
  Conf
  \big(
    \mathbb{R}^V,
    X
  \big)
  \underoverset{\simeq}{\;cc\;}{\longrightarrow}
  \Omega^{V+1} \Sigma^{V+1} X
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{RourkeSanderson00}{Rourke-Sanderson 00, Theorem 1, Theorem 2})

More generally:

\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 [[Cohomotopy charge map]] constitutes a [[homotopy equivalence]]

\begin{displaymath}
Conf\left( 
    \mathbb{R}^D, Y
  \right)
  \overset{cc}{\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{cc}{\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 [[Cohomotopy charge 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 cc}{\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{relation_to_classifying_space_of_the_symmetric_group}{}\paragraph*{{Relation to classifying space of the symmetric group}}\label{relation_to_classifying_space_of_the_symmetric_group}

Let $X= \mathbb{R}^\infty$. Then

\begin{itemize}%
\item the \emph{unordered} configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the [[classifying space]] $B \Sigma(n)$ of the [[symmetric group]] $\Sigma(n)$;

(e.g. \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 10})


\item the \emph{ordered} configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-[[action]], is a model for the $\Sigma(n)$-[[universal principal bundle]].



\end{itemize}
$\,$

\hypertarget{relation_to_james_construction}{}\paragraph*{{Relation to James construction}}\label{relation_to_james_construction}

The [[James construction]] of $X$ is [[homotopy equivalence|homotopy equivalent]] to the [[configuration space of points]] $C(\mathbb{R}^1, X)$ of points in the [[real line]] with labels taking values in $X$.

(e.g. \hyperlink{Boedigheimer87}{Bödigheimer 87, Example 9})

$\,$

\hypertarget{RelationToTwistedCohomotopy}{}\paragraph*{{In twisted Cohomotopy}}\label{RelationToTwistedCohomotopy}

The May-Segal theorem \ref{ScanningMapEquivalenceOverCartesianSpace} generalizes from [[Euclidean space]] to [[closed manifold|closed]] [[smooth manifolds]] if at the same time one passes from plain [[Cohomotopy]] to [[twisted Cohomotopy]], twisted, via the [[J-homomorphism]], by the [[tangent bundle]]:

\begin{prop}
\label{ScanningMapEquivalenceOverClosedManifold}\hypertarget{ScanningMapEquivalenceOverClosedManifold}{}
Let

\begin{enumerate}%
\item $X^n$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] $n$;


\item $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].



\end{enumerate}
Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

\begin{displaymath}
\underset{
    \color{blue}
    { \phantom{a} \atop \text{ J-twisted Cohomotopy space}}
  }{
    Maps_{{}_{/B O(n)}}
    \Big(
      X^n 
      \;,\; 
      S^{
        \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}}    
      }
      \!\sslash\!
      O(n)
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop 
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
\end{displaymath}
between

\begin{enumerate}%
\item the [[J-homomorphism|J]]-[[twisted Cohomotopy|twisted (n+k)-Cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$


\item the [[configuration space of points]] on $X^n$ with labels in $S^k$.



\end{enumerate}
\end{prop}
(\hyperlink{Boedigheimer87}{Bödigheimer 87, Prop. 2}, following \hyperlink{McDuff75}{McDuff 75})

\begin{prop}
\label{ScanningMapEquivalenceOverClosedFramedManifold}\hypertarget{ScanningMapEquivalenceOverClosedFramedManifold}{}
In the special case that the [[closed manifold]] $X^n$ in Prop. \ref{ScanningMapEquivalenceOverClosedManifold} is [[parallelizable manifold|parallelizable]], hence that its [[tangent bundle]] is [[trivial bundle|trivializable]], the statement of Prop. \ref{ScanningMapEquivalenceOverClosedManifold} reduces to this:

Let

\begin{enumerate}%
\item $X^n$ be a [[parallelizable manifold|parallelizable]] [[closed manifold]] of [[dimension]] $n$;


\item $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].



\end{enumerate}
Then the [[Cohomotopy charge map]] constitutes a [[weak homotopy equivalence]]

\begin{displaymath}
\underset{
    \color{blue}
    { \phantom{a} \atop \text{ Cohomotopy space}}
  }{
    Maps
    \Big(
      X^n 
      \;,\; 
      S^{
       n + k    
      }
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{Cohomotopy charge map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop 
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
\end{displaymath}
between

\begin{enumerate}%
\item $(n+k)$-[[Cohomotopy]] space of $X^n$, hence the [[space of maps]] from $X$ to the [[n-sphere|(n+k)-sphere]]


\item the [[configuration space of points]] on $X^n$ with labels in $S^k$.



\end{enumerate}
\end{prop}
\hypertarget{action_by_little_disk_operad_and_by_goodwillie_derivatives}{}\subsubsection*{{Action by little $n$-disk operad and by Goodwillie derivatives}}\label{action_by_little_disk_operad_and_by_goodwillie_derivatives}

Under some conditions and with suitable degrees/shifts, configuration spaces of points canonically have the [[structure]] of [[algebras over an operad]] over the [[little n-disk operad]] and the [[Goodwillie derivatives of the identitity functor]].

For more see \href{Goodwillie+derivatives+of+the+identity+functor#Properties}{there}

$\backslash$linebreak

\hypertarget{HomologyAndStabilization}{}\subsubsection*{{Homology and stabilization in homology}}\label{HomologyAndStabilization}

Let $X$ be a [[topological space]] which is the [[interior]] of a [[compact topological space|compact]] [[manifold with boundary]] $\overline{X}$. We may think of the [[boundary]] $\partial \overline X$ as consisting of the ``points at infinity'' in $X$.

In particular, there are then inclusion maps

\begin{equation}
Conf_n
  \big(
    X
  \big)
  \overset{i_n}{\longrightarrow}
  Conf_{n+1}
  \big(
    X
  \big)
\label{InclusionOfUnorderedConfigurationSpaces}\end{equation}
of the unordered configuration space of $n$ points in $X$ (Def. \ref{UnorderedUnlabeledConfigurations}) into that of $n + 1$ points, formalizing the idea of ``adding a point at infinity'' to a configuration. More formally, these maps are given by pushing configuration points away from the boundary a little and then adding a new point near to a point on the boundary of $X$.

(\hyperlink{RandalWilliams13}{Randal-Williams 13, Section 4})

The [[homotopy class]] of these maps depends (just) on the [[connected component]] of the [[boundary]] $\partial \overline{X}$ at which one chooses to bring in the new point. But for any choice, they have the following effect on [[cycles]] in [[ordinary homology]]:

\begin{prop}
\label{HomologicalStabilizationForUnorderedConfigurationSpaces}\hypertarget{HomologicalStabilizationForUnorderedConfigurationSpaces}{}
\textbf{(homological stabilization for unordered configuration spaces)}

Let $X$ be

\begin{itemize}%
\item a [[connected topological space|connected]] [[smooth manifold]]


\item which is the [[interior]] of a [[compact topological space|compact]] [[manifold with boundary]]


\item of [[dimension]] $dim(X) \geq 2$.



\end{itemize}
Then for all $n \in \mathbb{N}$ the inclusion [[maps]] \eqref{InclusionOfUnorderedConfigurationSpaces} are such that on [[ordinary homology]] with [[integer]] [[coefficients]] these maps induce [[split monomorphisms]] in all degrees,

\begin{displaymath}
H_\bullet
  \big( 
    Conf_n(X)
    ,
    \mathbb{Z}
  \big)
  \overset{
    H_\bullet( i_n, \mathbb{Z} )
  }{\hookrightarrow}
  H_\bullet
  \big( 
    Conf_{n+1}(X)
    ,
    \mathbb{Z}
  \big)
\end{displaymath}
and in degrees $\leq n/2$ these are even [[isomorphisms]]

\begin{displaymath}
H_p
  \big( 
    Conf_n(X)
    ,
    \mathbb{Z}
  \big)
  \underoverset{\simeq}{
    H_p( i_n, \mathbb{Z} )
  }{\hookrightarrow}
  H_p
  \big( 
    Conf_{n+1}(X)
    ,
    \mathbb{Z}
  \big)
  \phantom{AAAA}
  \text{for}
  \;
  p \leq n/2
  \,.
\end{displaymath}
Finally, for [[ordinary homology]] with [[rational number|rational]] [[coefficients]], these maps induce [[isomorphisms]] all the way up to degree $n$:

\begin{displaymath}
H_p
  \big( 
    Conf_n(X)
    ,
    \mathbb{Q}
  \big)
  \underoverset{\simeq}{
    H_p( i_n, \mathbb{Q} )
  }{\hookrightarrow}
  H_p
  \big( 
    Conf_{n+1}(X)
    ,
    \mathbb{Q}
  \big)
  \phantom{AAAA}
  \text{for}
  \;
  p \leq n
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{RandalWilliams13}{Randal-Williams 13, Theorem A and Threorem B})

$\backslash$linebreak

\hypertarget{RationalHomotopyType}{}\subsubsection*{{Rational homotopy type}}\label{RationalHomotopyType}

We discuss aspects of the [[rational homotopy type]] of configuration spaces of points. See also at \emph{[[graph complex]]}.

\hypertarget{Cohomology}{}\paragraph*{{Rational cohomology}}\label{Cohomology}

\begin{prop}
\label{RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}\hypertarget{RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}{}
\textbf{([[real cohomology]] of configuration spaces of ordered points in [[Euclidean space]])}

The [[real cohomology|real]] [[cohomology ring]] of the configuration spaces $\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$ (Def. \ref{OrderedUnlabeledConfigurations}) of $n$ ordered unlabeled points in [[Euclidean space]] $\mathbb{R}^D$

is [[generators and relations|generated]] by elements in degree $D-1$

\begin{displaymath}
\omega_{i j}
  \;\;
  \in
  H^2
  \Big(
    \underset{
      {}^{\{1, \cdots, n\}}
    }{
      Conf
    }
    \big( 
      \mathbb{R}^D
    \big),
    \mathbb{R}
  \Big)
\end{displaymath}
for $i, j \in \{1, \cdots, n\}$

subject to these three [[generators and relations|relations]]:

\begin{enumerate}%
\item \textbf{(anti-)symmetry)}

\begin{displaymath}
\omega_{i j} = (-1)^D \omega_{j i}
\end{displaymath}

\item \textbf{nilpotency}

\begin{displaymath}
\omega_{i j} \wedge \omega_{i j} \;=\; 0
\end{displaymath}

\item \textbf{3-term relation}

\begin{displaymath}
\omega_{i j} 
    \wedge 
  \omega_{j k} 
    + 
  \omega_{j k}
    \wedge 
  \omega_{k i} 
    + 
  \omega_{k i} 
    \wedge 
  \omega_{i j} 
    = 
  0
\end{displaymath}


\end{enumerate}
Hence:

\begin{equation}
H^\bullet
  \Big(
    \underset{
      {}^{\{1,\cdots,n\}}
    }{Conf}
    \big( 
      \mathbb{R}^D 
    \big),
    \mathbb{R}
  \Big)
  \;\simeq\;
  \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big]
  \Big/
  \left(
    \itexarray{
      \omega_{i j} = (-1)^D \omega_{j i}
      \\
      \omega_{i j} \wedge \omega_{i j} = 0
      \\
      \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0    
     }
     \;\;
     \text{for}\;
     i,j \in \{1, \cdots, n\}
  \right)
\label{RealCohomologyOfUnorderedConfigurationSpaceInEuclideanSpace}\end{equation}
\end{prop}
This is due to \hyperlink{Cohen76}{Cohen 76}, following \hyperlink{Arnold69}{Arnold 69}, \hyperlink{Cohen73}{Cohen 73}. See also \hyperlink{FelixTanre03}{Félix-Tanré 03, Section 2} \hyperlink{LambrechtsTourtchine09}{Lambrechts-Tourtchine 09, Section 3}.

See also at \emph{[[Fulton-MacPherson compactification]]} the section \emph{\href{Fulton-MacPherson+operad#deRhamCohomology}{de Rham cohomology}}.

\begin{remark}
\label{RealCohomologyOfConfigurationSpaceInTermsOfGraphCohomology}\hypertarget{RealCohomologyOfConfigurationSpaceInTermsOfGraphCohomology}{}
\textbf{([[real cohomology]] of the configuration space in terms of [[graph cohomology]])}

In the [[graph complex]]-model for the [[rational homotopy type]] of the ordered unlabled [[configuration space of points]] $\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$ the three relations in Prop. \ref{RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace} are incarnated as follows:

\begin{enumerate}%
\item a graph changes sign when one of its edges is reversed (\href{graph+complex#SignRulesForGraphs}{this Def.})


\item a graph with [[parallel edges]] is a vanishing graph (\href{graph+complex#VanishingGraphs}{this Def.})


\item the graph coboundary of a single trivalent internal vertex (\href{graph+complex#ThreeTermRelation}{this Example}).



\end{enumerate}
\end{remark}
$\backslash$linebreak

\hypertarget{RationalHomotopyGroups}{}\paragraph*{{Rational homotopy and Whitehead products}}\label{RationalHomotopyGroups}

Write again

\begin{displaymath}
Conf_n\big( \mathbb{R}^D \big) 
  \;\coloneqq\;
  \big( \mathbb{R}^D \big)^n \setminus FatDiag
\end{displaymath}
for the configuration space of $n$ ordered points in [[Euclidean space]].

\begin{prop}
\label{RationalHomotopyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}\hypertarget{RationalHomotopyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}{}
The [[Whitehead product]] [[super Lie algebra]] of [[rationalization|rationalized]] [[homotopy groups]]

\begin{displaymath}
L^n
  \;\coloneqq\;
  \pi_{\bullet+1}\Big(  Conf_n\big( \mathbb{R}^D \big) \Big)
  \otimes_{\mathbb{Z}} \mathbb{Q}
\end{displaymath}
is [[generators and relations|generated]] from elements

\begin{displaymath}
\omega^{i j}
  \;\in\;
  \pi_2
  \Big(
    Conf_n\big( \mathbb{R}^D \big)  
  \Big)
  \otimes_{\mathbb{Z}} \mathbb{Q}
  \phantom{AAA}
  i \neq j \in \{1, \cdots, n\}
  \,,
\end{displaymath}
subject to the following [[generators and relations|relations]]:

\begin{enumerate}%
\item $\omega^{i j} = (-1)^D \omega^{j i}$


\item $\big[ \omega^{i j}, \omega^{k l} \big]$ $\;\;\;$ if $i,j,k,l$ are pairwise distinct;


\item $\big[ \omega^{i j}, \omega^{j k} + \omega^{k i}  \big] = 0$.



\end{enumerate}
\end{prop}
This is due to \hyperlink{Kohno02}{Kohno 02}. See also \hyperlink{LambrechtsTourtchine09}{Lambrechts-Tourtchine 09, Section 3}.

$\backslash$linebreak

$\backslash$linebreak

\hypertarget{OccurrencesAndApplications}{}\subsection*{{Occurrences and Applications}}\label{OccurrencesAndApplications}

\hypertarget{compactification}{}\subsubsection*{{Compactification}}\label{compactification}

The [[Fulton-MacPherson compactification]] of configuration spaces of points in $\mathbb{R}^d$ serves to exhibit them as models for the [[little n-disk operad]].

\hypertarget{stable_splitting_of_mapping_spaces}{}\subsubsection*{{Stable splitting of mapping spaces}}\label{stable_splitting_of_mapping_spaces}

The [[stable splitting of mapping spaces]] says that [[suspension spectra]] of suitable [[mapping spaces]] are equivalently [[wedge sums]] of [[suspension spectra]] of configuration spaces of points.

\hypertarget{correlators_as_differential_forms_on_configuration_spaces}{}\subsubsection*{{Correlators as differential forms on configuration spaces}}\label{correlators_as_differential_forms_on_configuration_spaces}

In [[Euclidean field theory]] the [[correlators]] are often regarded as [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]]. Hence they become [[non-singular distributions]] after [[restriction of distributions]] to the corresponding configuration space of points.

For more on this see at \emph{[[correlators as differential forms on configuration spaces of points]]}.

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

\begin{itemize}%
\item [[Hilbert scheme]]

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

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

General accounts:

\begin{itemize}%
\item [[Edward Fadell]], [[Lee Neuwirth]], \emph{Configuration spaces}, Math. Scand. \textbf{10} (1962) 111-118, \href{http://www.ams.org/mathscinet-getitem?mr=141126}{MR141126}, \href{http://www.mscand.dk/article.php?id=1623}{pdf}


\item [[Edward Fadell]], [[Sufian Husseini]], \emph{Geometry and topology of configuration spaces}, Springer Monographs in Mathematics (2001), \href{http://www.ams.org/mathscinet-getitem?mr=2002k:55038}{MR2002k:55038}, xvi+313


\item [[Craig Westerland]], \emph{Configuration spaces in geometry and topology}, 2011 (\href{https://www.austms.org.au/Publ/Gazette/2011/Nov11/TechPaperWesterland.pdf}{pdf})

(in [[geometry]] and [[topology]])


\item [[Ben Knudsen]], \emph{Configuration spaces in algebraic topology} (\href{https://arxiv.org/abs/1803.11165}{arXiv:1803.11165})

(in [[algebraic topology]])



\end{itemize}
In relation to the [[space of finite subsets]]:

\begin{itemize}%
\item David Handel, \emph{Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces}, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (\href{https://www.math.uh.edu/~hjm/Vol26-4.html}{hjm:Vol26-4})


\item [[Yves Félix]], [[Daniel Tanré]] \emph{Rational homotopy of symmetric products and Spaces of finite subsets}, Contemp. Math 519 (2010): 77-92 (\href{http://tanre.org/Pro/Articles_files/SpnFiniteDef.pdf}{pdf})



\end{itemize}
The [[algebra over an operad|algebra]]-[[structure]] of configuration spaces over [[little n-disk operads]]/[[Fulton-MacPherson operads]]:

\begin{itemize}%
\item [[Martin Markl]], \emph{A compactification of the real configuration space as an operadic completion, J. Algebra 215 (1999), no. 1, 185–204}

\end{itemize}
\hypertarget{cohomotopy_charge_map_2}{}\subsubsection*{{Cohomotopy charge map}}\label{cohomotopy_charge_map_2}

The [[Cohomotopy charge map]] (``electric field map'', ``scanning map'') and hence the relation of configuration spaces to [[Cohomotopy]] goes back to

\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}) c Generalization of these constructions and results is due to


\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]])


\item Richard Manthorpe, [[Ulrike Tillmann]], \emph{Tubular configurations: equivariant scanning and splitting}, Journal of the London Mathematical Society, Volume 90, Issue 3 (\href{https://arxiv.org/abs/1307.5669}{arxiv:1307.5669}, \href{https://doi.org/10.1112/jlms/jdu050}{doi:10.1112/jlms/jdu050})



\end{itemize}
and generalization to [[equivariant homotopy theory]] is discussed in

\begin{itemize}%
\item [[Colin Rourke]], [[Brian Sanderson]], \emph{Equivariant Configuration Spaces}, J. London Math. Soc. 62 (2000) 544-552 (\href{https://arxiv.org/abs/math/9712216}{arXiv:math/9712216})

\end{itemize}
The relevant construction for the [[group completion]] of the configuration space

\begin{itemize}%
\item \hyperlink{Segal73}{Segal 73, Theorem 1}


\item [[Paolo Salvatore]], \emph{Configuration spaces with summable labels}, In: Aguadé J., Broto C., [[Carles Casacuberta]] (eds.) \emph{Cohomological Methods in Homotopy Theory}. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 (\href{https://arxiv.org/abs/math/9907073}{arXiv:math/9907073})



\end{itemize}
On the [[homotopy type]] of the [[space of maps|space of]] [[rational functions]] from the [[Riemann sphere]] to itself (related to the [[moduli space of monopoles]] in $\mathbb{R}^3$ and to the [[configuration space of points]] in $\mathbb{R}^2$):

\begin{itemize}%
\item [[Graeme Segal]], \emph{The topology of spaces of rational functions}, Acta Math. Volume 143 (1979), 39-72 (\href{https://projecteuclid.org/euclid.acta/1485890033}{euclid:1485890033})

\end{itemize}
See also

\begin{itemize}%
\item [[Sadok Kallel]], \emph{Particle Spaces on Manifolds and Generalized Poincaré Dualities} (\href{https://arxiv.org/abs/math/9810067}{arXiv:math/9810067})


\item Shingo Okuyama, Kazuhisa Shimakawa, \emph{Interactions of strings and equivariant homology theories}, (\href{https://arxiv.org/abs/0903.4667}{arXiv:0903.4667})



\end{itemize}
For relation to [[instantons]] via [[topological Yang-Mills theory]]:

\begin{itemize}%
\item [[Michael Atiyah]], [[John David Stuart Jones]], \emph{Topological aspects of Yang-Mills theory}, Comm. Math. Phys. Volume 61, Number 2 (1978), 97-118 (\href{https://projecteuclid.org/euclid.cmp/1103904210}{arXiv:1103904210})

\end{itemize}
In speculation regarding [[Galois theory]] over the [[sphere spectrum]]:

\begin{itemize}%
\item [[Jack Morava]], [[Jonathan Beardsley]], \emph{Toward a Galois theory of the integers over the sphere spectrum}, Journal of Geometry and Physics Volume 131, September 2018, Pages 41-51 (\href{https://arxiv.org/abs/1710.05992}{arXiv:1710.05992})

\end{itemize}
\hypertarget{stable_splitting_of_mapping_spaces_2}{}\subsubsection*{{Stable splitting of mapping spaces}}\label{stable_splitting_of_mapping_spaces_2}

The appearance of configuration spaces as summands in [[stable splittings of mapping spaces]] is originally due to

\begin{itemize}%
\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})

\end{itemize}
An alternative proof is due to

\begin{itemize}%
\item [[Ralph Cohen]], \emph{Stable proof of stable splittings}, Math. Proc. Camb. Phil. Soc., 1980, 88, 149 (\href{https://doi.org/10.1017/S030500410005742X}{doi:10.1017/S030500410005742X}, \href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/247D9F951F8AB99000E4FF6CB2DB9EA2/S030500410005742Xa.pdf/div-class-title-stable-proofs-of-stable-splittings-div.pdf}{pdf})

\end{itemize}
Review and generalization is in

\begin{itemize}%
\item \hyperlink{Boedigheimer87}{Boedigheimer 87}

\end{itemize}
and the relation to the [[Goodwillie-Taylor tower]] of mapping spaces is pointed out in

\begin{itemize}%
\item \hyperlink{Arone99}{Arone 99}

\end{itemize}
\hypertarget{ReferencesInGoodwillieCalculus}{}\subsubsection*{{In Goodwillie-calculus}}\label{ReferencesInGoodwillieCalculus}

The configuration spaces of a space $X$ appear as the [[Goodwillie derivatives]] of its [[mapping space]]/[[nonabelian cohomology]]-[[functor]] $Maps(X,-)$:

\begin{itemize}%
\item [[Greg Arone]], \emph{A generalization of Snaith-type filtration}, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (\href{https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf}{pdf})


\item [[Michael Ching]], \emph{Calculus of Functors and Configuration Spaces}, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (\href{https://www3.amherst.edu/~mching/Work/skye.pdf}{pdf})



\end{itemize}
\hypertarget{ReferencesCompactification}{}\subsubsection*{{Compactification}}\label{ReferencesCompactification}

A [[compactification]] of configuration spaces of points was introduced in

\begin{itemize}%
\item [[William Fulton]], [[Robert MacPherson]], \emph{A compactification of configuration spaces}, Ann. of Math. (2), 139(1):183–225, 1994.

\end{itemize}
and an [[operad]]-[[structure]] defined on it ([[Fulton-MacPherson operad]]) in

\begin{itemize}%
\item [[Ezra Getzler]], [[John Jones]], \emph{Operads, homotopy algebra and iterated integrals for double loop spaces} (\href{https://arxiv.org/abs/hep-th/9403055}{arXiv:hep-th/9403055})

\end{itemize}
Review includes

\begin{itemize}%
\item [[Pascal Lambrechts]], [[Ismar Volić]], section 5 of \emph{Formality of the little N-disks operad}, Memoirs of the American Mathematical Society ; no. 1079, 2014 (\href{https://arxiv.org/abs/0808.0457}{arXiv``0808.0457}, \href{http://dx.doi.org/10.1090/memo/1079}{doi:10.1090/memo/1079})

\end{itemize}
This underlies the models of configuration spaces by [[graph complexes]], see there for more.

\hypertarget{ReferencesCohomology}{}\subsubsection*{{Homology and cohomology}}\label{ReferencesCohomology}

General discussion of [[ordinary homology]]/[[ordinary cohomology]] of configuration spaces of points:

\begin{itemize}%
\item [[Vladimir Arnold]], \emph{The cohomology ring of the colored braid group}, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 (\href{http://mi.mathnet.ru/eng/mz6827}{mathnet:mz6827})


\item [[Fred Cohen]], \emph{Cohomology of braid spaces}, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 763-766 (\href{https://projecteuclid.org/euclid.bams/1183534761}{euclid:1183534761})


\item [[Fred Cohen]], \emph{The homology of $C_{n+1}$-Spaces, $n \geq 0$}, In: \emph{The Homology of Iterated Loop Spaces}, Lecture Notes in Mathematics, vol 533. Springer 1976(\href{https://doi.org/10.1007/BFb0080467}{doi:10.1007/BFb0080467})


\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})


\item E. Ossa, \emph{On the cohomology of configuration spaces}, In: Broto C., [[Carles Casacuberta]], Mislin G. (eds.), \emph{Algebraic Topology: New Trends in Localization and Periodicity}, Progress in Mathematics, vol 136. Birkhäuser Basel (1996) (\href{https://doi.org/10.1007/978-3-0348-9018-2_26}{doi:10.1007/978-3-0348-9018-2\_26})


\item [[Yves Félix]], [[Jean-Claude Thomas]], \emph{Rational Betti numbers of configuration spaces}, Topology and its Applications, Volume 102, Issue 2, 8 April 2000, Pages 139-149 ()


\item [[Oscar Randal-Williams]], \emph{Homological stability for unordered configuration spaces}, The Quarterly Journal of Mathematics, Volume 64, Issue 1, March 2013, Pages 303–326 (\href{https://arxiv.org/abs/1105.5257}{arXiv:1105.5257})


\item [[Yves Félix]], [[Daniel Tanré]], \emph{The cohomology algebra of unordered configuration spaces}, Journal of the LMS, Vol 72, Issue 2 (\href{https://arxiv.org/abs/math/0311323}{arxiv:math/0311323}, \href{https://doi.org/10.1112/S0024610705006794}{doi:10.1112/S0024610705006794})


\item Thomas Church, \emph{Homological stability for configuration spaces of manifolds} (\href{https://arxiv.org/abs/1602.04748}{arxiv:1602.04748})


\item [[Ben Knudsen]], \emph{Betti numbers and stability for configuration spaces via factorization homology}, Algebr. Geom. Topol. 17 (2017) 3137-3187 (\href{https://arxiv.org/abs/1405.6696}{arXiv:1405.6696})


\item Christoph Schiessl, \emph{Betti numbers of unordered configuration spaces of the torus} (\href{https://arxiv.org/abs/1602.04748}{arxiv:1602.04748})


\item Christoph Schiessl, \emph{Integral cohomology of configuration spaces of the sphere} (\href{https://arxiv.org/abs/1801.04273}{arxiv:1801.04273})


\item Dan Petersen, \emph{Cohomology of generalized configuration spaces} (\href{https://arxiv.org/abs/1807.07293}{arXiv:1807.07293})


\item Roberto Pagaria, \emph{The cohomology rings of the unordered configuration spaces of the torus} (\href{https://arxiv.org/abs/1901.01171}{arxiv:1901.01171})



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

Discussion of [[homotopy groups]] of configuration spaces:

\begin{itemize}%
\item \hyperlink{Kohno02}{Kohno 02}


\item [[Pascal Lambrechts]], [[Victor Tourtchine]], \emph{Homotopy graph-complex for configuration and knot spaces}, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 (\href{https://arxiv.org/abs/math/0611766}{arxiv:math/0611766})


\item [[Sadok Kallel]], Ines Saihi, \emph{Homotopy Groups of Diagonal Complements}, Algebr. Geom. Topol. 16 (2016) 2949-2980 (\href{https://arxiv.org/abs/1306.6272}{arXiv:1306.6272})



\end{itemize}
\hypertarget{rational_homotopy_type_2}{}\subsubsection*{{Rational homotopy type}}\label{rational_homotopy_type_2}

Discussion of the [[rational homotopy type]]:

\begin{itemize}%
\item [[Igor Kriz]], \emph{On the Rational Homotopy Type of Configuration Spaces}, Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 (\href{https://www.jstor.org/stable/2946581}{jstor:2946581})

\end{itemize}
\hypertarget{cohomology_modeled_by_graph_complexes}{}\subsubsection*{{Cohomology modeled by graph complexes}}\label{cohomology_modeled_by_graph_complexes}

That the [[de Rham cohomology]] of (the [[Fulton-MacPherson compactification]] of) configuration spaces of points may be modeled by [[graph complexes]] (exhibiting [[formality of the little n-disk operad]]) is due to

\begin{itemize}%
\item [[Maxim Kontsevich]], around Def. 15 and Lemma 3 in \emph{Operads and Motives in Deformation Quantization}, Lett.Math.Phys.48:35-72,1999 (\href{https://arxiv.org/abs/math/9904055}{arXiv:math/9904055})

\end{itemize}
nicely reviewed in \hyperlink{LambrechtsVolic14}{Lambrechts-Volic 14}

Further discussion of the graph complex as a model for the [[de Rham cohomology]] of [[configuration spaces of points]] is in

\begin{itemize}%
\item Najib Idrissi, \emph{The Lambrechts-Stanley Model of Configuration Spaces}, Invent. Math, 2018 (\href{https://arxiv.org/abs/1608.08054}{arXiv:1608.08054}, \href{https://dx.doi.org/10.1007/s00222-018-0842-9}{doi:10.1007/s00222-018-0842-9})


\item [[Ricardo Campos]], [[Thomas Willwacher]], \emph{A model for configuration spaces of points} (\href{https://arxiv.org/abs/1604.02043}{arXiv:1604.02043})


\item [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], \emph{Configuration Spaces of Manifolds with Boundary} (\href{https://arxiv.org/abs/1802.00716}{arXiv:1802.00716})


\item [[Ricardo Campos]], Julien Ducoulombier, Najib Idrissi, [[Thomas Willwacher]], \emph{A model for framed configuration spaces of points} (\href{https://arxiv.org/abs/1807.08319}{arXiv:1807.08319})



\end{itemize}
\hypertarget{ReferencesLoopSpacesOfConfigurationSpaces}{}\subsubsection*{{Loop spaces of configuration spaces of points}}\label{ReferencesLoopSpacesOfConfigurationSpaces}

On [[loop spaces]] of [[configuration spaces of points]]:

\begin{itemize}%
\item [[Edward Fadell]], [[Sufian Husseini]], \emph{The space of loops on configuration spaces and the Majer-Terracini index}, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 (\href{https://projecteuclid.org/euclid.tmna/1476842829}{euclid:tmna/1476842829})


\item [[Fred Cohen]], [[Samuel Gitler]], \emph{Loop spaces of configuration spaces, braid-like groups, and knots}, In: Aguadé J., Broto C., [[Carles Casacuberta]] (eds.) \emph{Cohomological Methods in Homotopy Theory}. Progress in Mathematics, vol 196. Birkhäuser, Basel (\href{https://doi.org/10.1007/978-3-0348-8312-2_7}{doi:10.1007/978-3-0348-8312-2\_7})


\item [[Toshitake Kohno]], \emph{Loop spaces of configuration spaces and finite type invariants}, Geom. Topol. Monogr. 4 (2002) 143-160 (\href{https://arxiv.org/abs/math/0211056}{arXiv:math/0211056})


\item [[Fred Cohen]], [[Samuel Gitler]], \emph{On loop spaces of configuration spaces}, Trans. Amer. Math. Soc. \textbf{354} (2002), no. 5, 1705--1748, (\href{https://www.jstor.org/stable/2693715}{jstor:2693715}, \href{http://www.ams.org/mathscinet-getitem?mr=1881013}{MR2002m:55020})

(on [[ordinary homology]] of [[loop spaces]] of configuration spaces)



\end{itemize}
\hypertarget{AsModuliOfD0D4BraneBoundStates}{}\subsubsection*{{As moduli of Dp-D(p+4)-brane bound states:}}\label{AsModuliOfD0D4BraneBoundStates}

Discussion of [[configuration spaces of points]] as [[moduli spaces]] of [[D0-D4-brane bound states]]

\begin{itemize}%
\item [[Cumrun Vafa]], \emph{Instantons on D-branes}, Nucl. Phys. B463 (1996) 435-442 (\href{https://arxiv.org/abs/hep-th/9512078}{arXiv:hep-th/9512078})

\end{itemize}
with emphasis to the resulting [[configuration spaces of points]], as in

\begin{itemize}%
\item [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: \emph{A Strong Coupling Test of S-Duality}, Nucl. Phys. B431:3-77, 1994 (\href{https://arxiv.org/abs/hep-th/9408074}{arXiv:hep-th/9408074})

\end{itemize}
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