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\begin{document}

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\section*{cohomotopy charge map}

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

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

[[!include homotopy - contents]]

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ForChargedPoints}{For charged points}\dotfill \pageref*{ForChargedPoints} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_cobordism_classes_by_their_cohomotopy_charge}{Characterization of cobordism classes by their Cohomotopy charge}\dotfill \pageref*{characterization_of_cobordism_classes_by_their_cohomotopy_charge} \linebreak
\noindent\hyperlink{CharacterizationOfPointConfigurations}{Characterization of point configurations by their Cohomotopy charge}\dotfill \pageref*{CharacterizationOfPointConfigurations} \linebreak
\noindent\hyperlink{on_euclidean_spaces_via_plain_cohomotopy}{On Euclidean spaces via plain Cohomotopy}\dotfill \pageref*{on_euclidean_spaces_via_plain_cohomotopy} \linebreak
\noindent\hyperlink{on_closed_manifolds_via_twisted_cohomotopy}{On closed manifolds via twisted Cohomotopy}\dotfill \pageref*{on_closed_manifolds_via_twisted_cohomotopy} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{for_point_configurations}{For point configurations}\dotfill \pageref*{for_point_configurations} \linebreak


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

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

The \emph{Cohomotopy charge map} is the [[function]] that assigns to a [[configuration space (mathematics)|configuration]] of [[normally framed submanifolds]] of [[codimension]] $n$ their total [[charge]] as measured in $n$-[[Cohomotopy]]-[[generalized cohomology|cohomology theory]].

Concretely, this is the function which assigns to a [[normally framed submanifold]] its \textbf{asymptotic normal distance function}, namely the [[distance]] from the [[submanifold]] measured

\begin{enumerate}%
\item in [[direction vector|direction]] [[orthogonality|perpendicular]] to the submanifold, as encoded by the [[normal framing]];


\item asymptotically, regarding all points outside a [[tubular neighbourhood]] as being [[one-point compactification|at infinity]].



\end{enumerate}
\begin{quote}%
graphics grabbed from \hyperlink{SatiSchreiber19}{SS 19}


\end{quote}
For general $n$ this is known as the ``[[Pontrjagin-Thom collapse construction]]''.

\hypertarget{ForChargedPoints}{}\subsubsection*{{For charged points}}\label{ForChargedPoints}

For maximal [[codimension]] $n$ inside an [[orientation|oriented]] [[smooth manifold|manifold]], hence for 0-dimensional submanifolds, hence for [[configuration space of points|configurations of points]] and with all points regarded as equipped with positive normal framing, the Cohomotopy charge map 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}, \hyperlink{ManthorpeTillmann13}{Manthorpe-Tillmann 13}):

In maximal [[codimension]] $D \in \mathbb{N}$, the \emph{Cohomotopy charge map} is thus 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}).

\begin{quote}%
graphics grabbed from \hyperlink{SatiSchreiber19}{SS 19}


\end{quote}
(See also at \emph{[[cobordism]] -- \href{cobordism#RelationToCohomotopy}{Relation to Cohomotopy}}.)

This 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 \hyperlink{SatiSchreiber19}{SS 19}


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

\hypertarget{characterization_of_cobordism_classes_by_their_cohomotopy_charge}{}\subsubsection*{{Characterization of cobordism classes by their Cohomotopy charge}}\label{characterization_of_cobordism_classes_by_their_cohomotopy_charge}

The unstable [[Pontrjagin-Thom theorem]] states that Cohomotopy charge faithfully reflects [[configuration space (mathematics)|configurations]] of [[normally framed submanifolds]] up to normally framed embedded [[cobordism]], hence that the [[Pontrjagin-Thom collapse construction]] induces a [[bijection]] between [[cobordism classes]] of [[normally framed submanifolds]] and the [[Cohomotopy set]] in degree the respective [[codimension]]:

\begin{displaymath}
\left\{
    {
      {
        \text{normally framed submanifolds}
      }
      \atop
      {
        \text{in}\;X\;\text{of codimension}\; n
      }
    }
  \right\}
  \Big/_{\sim_{cobordism}}
  \underoverset{\simeq}{cc}{\longrightarrow}
  \underset{
     \mathclap{
       \color{blue}
       {
         \text{Cohomotopy}
         \atop
         \text{set}
       }
     }
  }{  
    \pi^n\big( X \big)
  }
\end{displaymath}
For more details see \href{cohomotopy#RelationToCobordismGroup}{here}.

In goos situations this [[bijection]] of [[sets]] of [[homotopy classes]] enhances to a [[weak equivalence]] of [[configuration space (mathematics)|configuration spaces]]/[[cocycle spaces]]. See \emph{\hyperlink{CharacterizationOfPointConfigurations}{Characterization of point configurations by their Cohomotopy charge}} below.

$\backslash$linebreak

\hypertarget{CharacterizationOfPointConfigurations}{}\subsubsection*{{Characterization of point configurations by their Cohomotopy charge}}\label{CharacterizationOfPointConfigurations}

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]].

\hypertarget{on_euclidean_spaces_via_plain_cohomotopy}{}\paragraph*{{On Euclidean spaces via plain Cohomotopy}}\label{on_euclidean_spaces_via_plain_cohomotopy}

\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 (\href{configuration+space+of+points#eq:UnorderedUnlabeledConfigurationSpace}{here}) of [[Euclidean space]] $\mathbb{R}^D$ to the $D$-fold [[iterated based loop space]] of the [[n-sphere|D-sphere]], exhibits the [[group completion]] (\href{configuration+space+of+points#eq:GroupCompletionOfConfigurationSpaceMonoid}{here}) 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}
\end{displaymath}
is a [[weak homotopy equivalence]] from the configuration space (\href{configuration+space+of+points#eq:UnorderedLabeledCOnfigurationSpace}{here}) of unordered points with labels in $S^k$ and vanishing at the base point of the label space to the $D$-fold [[iterated loop space]] of the [[n-sphere|D+k-sphere]].

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

\hypertarget{on_closed_manifolds_via_twisted_cohomotopy}{}\paragraph*{{On closed manifolds via twisted Cohomotopy}}\label{on_closed_manifolds_via_twisted_cohomotopy}

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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

In the general guise of the [[Pontrjagin-Thom construction]] the concept of Cohomotopy charge goes back to

\begin{itemize}%
\item [[René Thom]], \emph{Quelques propri\'e{}t\'e{}s globales des vari\'e{}t\'e{}s diff\'e{}rentiables} Comment. Math. Helv. 28, (1954). 17-86 (\href{http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259}{digiz:GDZPPN002056259})


\item [[Lev Pontrjagin]], \emph{Smooth manifolds and their applications in Homotopy theory}, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/pont4.pdf}{pdf}, \href{https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001}{doi:10.1142/9789812772107\_0001})



\end{itemize}
A textbook account of the unstable [[Pontrjagin-Thom theorem]] is in

\begin{itemize}%
\item [[Antoni Kosinski]], chapter IX of \emph{Differential manifolds}, Academic Press 1993 (\href{http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf}{pdf})

\end{itemize}
\hypertarget{for_point_configurations}{}\subsubsection*{{For point configurations}}\label{for_point_configurations}

The theorem that, with due care, for [[configuration space of points|point configurations]] the Cohomotopy charge map is in fact a [[weak homotopy equivalence]] between the [[configuration space of points]] and the [[Cohomotopy]] [[cocycle space]] originates with

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



\end{itemize}
with comprehensive review in

\begin{itemize}%
\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}
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 [[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})


\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})


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


\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]} (\href{https://arxiv.org/abs/1909.12277}{arXiv:1909.12277})



\end{itemize}
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[[!redirects Cohomotopy charge]] [[!redirects Cohomotopy charges]]



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