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\begin{document}

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\section*{cohomotopy}

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

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

[[!include homotopy - contents]]

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

[[!include cohomology - contents]]

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

[[!include manifolds and cobordisms - 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{AsTheAbsoluteCohomologyTheory}{As the absolute cohomology theory}\dotfill \pageref*{AsTheAbsoluteCohomologyTheory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{hopf_degree_theorem}{Hopf degree theorem}\dotfill \pageref*{hopf_degree_theorem} \linebreak
\noindent\hyperlink{poincarhopf_theorem}{Poincaré–Hopf theorem}\dotfill \pageref*{poincarhopf_theorem} \linebreak
\noindent\hyperlink{relation_to_freudenthal_suspension_theorem}{Relation to Freudenthal suspension theorem}\dotfill \pageref*{relation_to_freudenthal_suspension_theorem} \linebreak
\noindent\hyperlink{smooth_representatives}{Smooth representatives}\dotfill \pageref*{smooth_representatives} \linebreak
\noindent\hyperlink{RelationToCobordismGroup}{PT-Construction and normally framed submanifolds}\dotfill \pageref*{RelationToCobordismGroup} \linebreak
\noindent\hyperlink{RelationToConfigurationSpaces}{Cohomotopy charge map and Relation to configuration spaces}\dotfill \pageref*{RelationToConfigurationSpaces} \linebreak
\noindent\hyperlink{complexrational_cohomotopy_and_moduli_space_of_yang0mills_monopoles}{Complex-rational Cohomotopy and moduli space of Yang-0Mills monopoles}\dotfill \pageref*{complexrational_cohomotopy_and_moduli_space_of_yang0mills_monopoles} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{On4Manifolds}{Of 4-Manifolds}\dotfill \pageref*{On4Manifolds} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{cohomotopy_cocycle_spaces}{Cohomotopy cocycle spaces}\dotfill \pageref*{cohomotopy_cocycle_spaces} \linebreak
\noindent\hyperlink{relation_to_knots_and_links}{Relation to knots and links}\dotfill \pageref*{relation_to_knots_and_links} \linebreak
\noindent\hyperlink{application_to_persistency}{Application to persistency}\dotfill \pageref*{application_to_persistency} \linebreak
\noindent\hyperlink{equivariant_cohomotopy}{Equivariant Cohomotopy}\dotfill \pageref*{equivariant_cohomotopy} \linebreak
\noindent\hyperlink{in_spectral_geometry}{In spectral geometry}\dotfill \pageref*{in_spectral_geometry} \linebreak


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

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

\emph{Cohomotopy cohomology theory} $\pi^\bullet$ is the ([[non-abelian cohomology|non-abelian]]) [[generalized cohomology theory]] whose [[cocycle spaces]] are [[spaces of maps]] into an [[n-sphere]], hence whose [[cohomology classes]] are [[homotopy classes]] of [[maps]] into an [[n-sphere]]:

\begin{displaymath}
\pi^n(X) \;\coloneqq\; Maps\big( X, S^n\big)/_{\sim_{homotopy}}
\end{displaymath}
So, dually to how [[homotopy groups]]

\begin{displaymath}
\pi_n(X)\coloneqq Maps\big( S^n, X\big)/_{\sim_{homotopy}}
\end{displaymath}
are [[groups]] of [[homotopy classes]] of maps \emph{out} of [[spheres]], Cohomotopy sets are sets of homotopy classes of maps \emph{into} spheres, whence the [[duality|dual]] name.

If instead one considers only the [[stable homotopy theory|stable]] aspect of Cohomotopy sets, by mapping into the [[stabilization]] of the spheres, hence into (some [[suspension spectrum|suspension]] of) the [[sphere spectrum]], then one speaks of \emph{[[stable Cohomotopy]]}, written

\begin{displaymath}
\mathbb{S}^n(X) 
    \;\coloneqq\; 
  Maps
  \big(
    \Sigma^\infty X , \Sigma^{\infty} S^n
  \big)/_{\sim_{homotopy}}
  \,.
\end{displaymath}
In other words, the [[generalized (Eilenberg-Steenrod) cohomology]] theory which is [[Brown representability theorem|represented]] by the [[sphere spectrum]] is \emph{[[stable Cohomotopy]]}.

\begin{remark}
\label{RemarkOnTerminology}\hypertarget{RemarkOnTerminology}{}
\textbf{(terminology)}

Therefore ``Cohomotopy theory'' is really shorthand for ``Cohomotopy cohomology theory'' and as such is [[duality|dual]] to \emph{homotopy homology theory}, which in the stable case is known as \emph{[[stable homotopy homology theory]]}.

In particular, cohomotopy theory is a [[concrete particular]] and not dual to the [[abstract general]] of [[homotopy theory]]; and is hence also not on par with the [[abstract general]] of [[cohomology theory]]. Rather, Cohomotopy theory is one \emph{instance} of a [[generalized cohomology|cohomologyy theory]], and as such is a sibling of [[ordinary cohomology theory]] ([[HR]]-theory)), [[K-theory]], etc.

To emphasize this, one might, in the [[stable cohomotopy theory|stable case]], say \emph{$\mathbb{S}$-theory} instead of ``stable Cohomotopy theory''; where $\mathbb{S}$ denotes the [[sphere spectrum]]. In the unstable case there is no widely adopted notation, but one might consider saying ``$\mathbf{\pi}$-theory'' (with $\pi$ the established symbol for (co)[[homotopy groups]]) or \emph{$S$-theory} (with ``$S$'' for [[n-spheres]] $S^n$) for unstable Cohomotopy theory.

In any case, to highlight that Cohomotopy theory is a [[concrete particular]] and not an [[abstract general]], it makes good sense to capitalize the term and speak of \emph{Cohomotopy cohomology theory} or just \emph{Cohomotopy theory}, for short.

The following table indicates the pattern:

\newline | [[ordinary cohomology]]:  [[HR]]-[[generalized (Eilenberg-Steenrod) cohomology theory|cohomology theory]] | [[ordinary homology]]: [[HR]]-[[generalized homology theory|homology theory]] | [[Eilenberg-MacLane space]] $K(R,n)$ | [[Eilenberg-MacLane spectrum]]  $H R$ | | [[topological K-theory|K-cohomology theory]] | [[K-homology theory]] | [[stable unitary group]]  [[stable unitary group|BU]] | [[K-theory spectrum]]  [[KU]], \ldots{} |

\end{remark}
As for any [[generalized cohomology theory]] there are immediate variants to plain Cohomotopy theory, as shown in the following table:

[[!include flavours of cohomotopy -- table]]

\hypertarget{AsTheAbsoluteCohomologyTheory}{}\subsubsection*{{As the absolute cohomology theory}}\label{AsTheAbsoluteCohomologyTheory}

In some sense Cohomotopy is the most fundamental of all [[generalized cohomology theories]] (``[[The Music of the Spheres]]'').

Concretely, [[stable Cohomotopy cohomology theory]] is the [[initial object]] among [[multiplicative cohomology theories]], in that the [[sphere spectrum]] is the [[initial object in an (infinity,1)-category|initial object]] in ([[E-infinity ring|E-infinity]]) [[ring spectra]]. This means that for any other [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology theory|cohomology theory]] $E$ there is an essentially unique multiplicative [[natural transformation]]

\begin{equation}
\mathbb{S}^\bullet(X)
  \overset{\beta(X)}{\longrightarrow}
  E^\bullet(X)
\label{BoardmanHomomorphism}\end{equation}
from Cohomotopy cohomology groups to $E$-cohomology groups -- the \emph{[[Boardman homomorphism]]}.

Specifically for $E = K \mathbb{F}$ the [[algebraic K-theory]] of a [[field]] $\mathbb{F}$ (such as a [[prime field]] $\mathbb{F}_p$) there is such a comparison morphism; and another way how stable Cohomotopy is the most fundamental of all K-theories is that it is equivalently the [[algebraic K-theory]] over the ``absolute base'', namely over the ``[[field with one element]]'' $\mathbb{F}_1$ (see \href{stable+cohomotopy#AsAlgebraicKTheoryOverTheFieldWithOneElement}{there} for more):

\begin{displaymath}
\mathbb{S}^\bullet(X) 
  \;\simeq\;
  K\mathbb{F}_1 ^ \bullet(X)
  \,.
\end{displaymath}
For example, for $E =$ [[KU]] and in the case of $G$-[[equivariant cohomology theory]] ([[equivariant Cohomotopy theory]] and [[equivariant K-theory]]) the [[Boardman homomorphism]] \eqref{BoardmanHomomorphism} gives the comparison map

\begin{displaymath}
\mathbb{S}^0_G(\ast)
  \simeq
  R_{\mathbb{F}_1}(G)
  \simeq
  A(G)
  \overset{\mathbb{C}[-]}{\longrightarrow}
  R_{\mathbb{C}}(G)
  \simeq
  KU^0_G(\ast)
\end{displaymath}
from the [[Burnside ring]] to the [[representation ring]] of the [[finite group]] $G$, by forming [[permutation representations]]; where we may think of the [[Burnside ring]] as being the [[representation ring]] over the ``[[field with one element]]'' (see e.g. \href{field+with+one+element#ChuLorscheidSanthanam10}{Chu-Lorscheid-Santhanam 10}), as indicated above.

So far, this applies to [[stable Cohomotopy theory]], which historically has received almost all the attention. But, while [[stabilization]] makes the immensely rich nature of [[homotopy theory]] a tad more tractable, it is only an approximation (just the first [[Goodwillie calculus|Goodwillie derivative]]!) of full unstable/[[non-abelian cohomology]]. Hence the one concept more fundamental than stable Cohomotopy theory is actual Cohomotopy theory.

For example, the classification of [[Yang-Mills instantons]] on $\mathbb{R}^4$ is typically regarded in the [[non-abelian cohomology]] theory represented by the [[classifying space]] $B SU(N)$ of the [[special unitary group]] (for $N \geq 2$, starting with [[SU(2)]])

\begin{displaymath}
(B SU(N))^0
  \Big(
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
  \Big)
  \;\coloneqq\;
  \Big\{ 
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
    \to
    B SU(N)
  \Big\}/_{\sim_{homotopy}}
  \;\simeq\;
  \underset{
    \mathclap{
    \color{blue}
    {\text{instanton} \atop \text{number}}
    }
  }{
    \mathbb{Z}
  }
  \,.
\end{displaymath}
But since the [[one-point compactification]] of 4d [[Euclidean space]] is the [[4-sphere]] $\big( \mathbb{R}^4\big)^{cpt} \simeq S^4$, this classification factors through one in unstable Cohomotopy theory, via the ``unstable Boardman homomorphism'' $S^4 \longrightarrow B SU(N)$ representing the generator of the 4th [[homotopy group]] of $B SU(N)$ (see \href{unitary+group#HomotopyGroups}{there})

\begin{displaymath}
\pi^0
  \Big(
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
  \Big)
  \;\coloneqq\;
  \Big\{ 
    \big(
      \mathbb{R}^{4}
    \big)^{cpt}
    \to
    S^4
  \Big\}/_{\sim_{homotopy}}
  \;\simeq\;
  \underset{
    \mathclap{
    \color{blue}
    {\text{instanton} \atop \text{number}}
    }
  }{
    \mathbb{Z}
  }
  \,.
\end{displaymath}
(see \hyperlink{SatiSchreiber19}{SS 19, p. 9-10}) This is the tip of an iceberg. Which needs to be discussed elsewhere.

$\backslash$linebreak

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

\hypertarget{hopf_degree_theorem}{}\subsubsection*{{Hopf degree theorem}}\label{hopf_degree_theorem}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[Hopf degree theorem]])}

Let $n \in \mathbb{N}$ be a [[natural number]] and $X \in Mfd$ be a [[connected topological space|connected]] [[orientation|orientable]] [[closed manifold]] of [[dimension]] $n$. Then the $n$th [[cohomotopy]] classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in [[bijection]] to the [[degree of a continuous function|degree]] $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

\begin{displaymath}
\pi^n(X) 
    \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow}
   H^n(X,\mathbb{Z}) 
     \;\simeq\;   
   \mathbb{Z}
\end{displaymath}
from $n$th [[cohomotopy]] to $n$th [[integral cohomology]] is a [[bijection]].

\end{prop}
(e.g. \hyperlink{Kosinski93}{Kosinski 93, IX (5.8)})

\hypertarget{poincarhopf_theorem}{}\subsubsection*{{Poincaré–Hopf theorem}}\label{poincarhopf_theorem}

\begin{itemize}%
\item [[Poincaré–Hopf theorem]]

\end{itemize}
\hypertarget{relation_to_freudenthal_suspension_theorem}{}\subsubsection*{{Relation to Freudenthal suspension theorem}}\label{relation_to_freudenthal_suspension_theorem}

relation to the [[Freudenthal suspension theorem]] (\hyperlink{Spanier49}{Spanier 49, section 9})

\hypertarget{smooth_representatives}{}\subsubsection*{{Smooth representatives}}\label{smooth_representatives}

For $X$ a [[compact topological space|compact]] [[smooth manifold]], there is a [[smooth function]] $X \to S^n$ representing every cohomotopy class (with respect to the standard [[smooth structure]] on the [[sphere]] [[manifold]]).

\hypertarget{RelationToCobordismGroup}{}\subsubsection*{{PT-Construction and normally framed submanifolds}}\label{RelationToCobordismGroup}

For $X$ a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] $D$, the assignment of [[Cohomotopy charge]] ([[Pontryagin-Thom construction]], e.g. \hyperlink{Kosinski93}{Kosinski 93, IX.5}) identifies the [[set]]

\begin{displaymath}
SubMfd_{/bord}^{d}(X)
\end{displaymath}
of [[cobordism classes]] of [[closed manifold|closed]] and [[normally framed submanifolds]] $\Sigma \overset{\iota}{\hookrightarrow} X$ of [[dimension]] $d$ inside $X$ with the [[cohomotopy]] $\pi^{D-d}(X)$ of $X$ in degree $D- d$

\begin{displaymath}
SubMfd_{/bord}^{d}(X)
    \underoverset{\simeq}{\;\;\;PT\;\;\;}{\longrightarrow}
  \pi^{D-d}(X)
  \,.
\end{displaymath}
(e.g. \hyperlink{Kosinski93}{Kosinski 93, IX Theorem (5.5)})

In particular, by this [[bijection]] the canonical [[group]] [[structure]] on [[cobordism groups]] in sufficiently high [[codimension]] (essentially given by [[disjoint union]] of [[submanifolds]]) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

$\,$

\begin{quote}%
graphics grabbed form \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
Here the [[normal framing]] of the submanifolds plays the role of the [[charge]] in [[Cohomotopy]] which they carry:

$\,$

\begin{quote}%
graphics grabbed form \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
For example:

$\,$

\begin{quote}%
graphics grabbed form \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
This construction generalizes to [[equivariant cohomotopy]], see \href{equivariant+cohomotopy#PontryaginThomConstruction}{there}.

With the [[equivariant Hopf degree theorem]] the above example has the following $\mathbb{Z}_2$-[[equivariant homotopy theory|equivariant]] version (see \href{equivariant+Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationSpheres}{there}):

$\,$

\begin{quote}%
graphics grabbed form \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
Further by the [[equivariant Hopf degree theorem]] (see \href{equivariant+Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationTori}{there}), this example generalizes to [[equivariant cohomotopy]] of [[toroidal orbifold|toroidal]] [[orientifolds]]:

$\,$

\begin{quote}%
graphics grabbed form \hyperlink{SatiSchreiber19}{Sati-Schreiber 19}


\end{quote}
$\backslash$linebreak

\hypertarget{RelationToConfigurationSpaces}{}\subsubsection*{{Cohomotopy charge map and Relation to configuration spaces}}\label{RelationToConfigurationSpaces}

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'' (\href{Cohomotopy+charge#Salvatore01}{Salvatore 01} following \href{Cohomotopy+charge#Segal73}{Segal 73, Section 1}, see also \href{Cohomotopy+charge#Knudsen18}{Knudsen 18, p. 49}) or the ``scanning map'' (\href{Cohomotopy+charge#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]] (\href{Cohomotopy+charge#Segal73}{Segal 73, Section 3}).

This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeleed 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}
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 (\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}
(\href{Cohomotopy+charge#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}
(\href{Cohomotopy+charge#Segal73}{Segal 73, Theorem 3})

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}
(\href{Cohomotopy+charge#Boedigheimer87}{Bödigheimer 87, Prop. 2}, following \href{Cohomotopy+charge#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}
$\backslash$linebreak

\hypertarget{complexrational_cohomotopy_and_moduli_space_of_yang0mills_monopoles}{}\subsubsection*{{Complex-rational Cohomotopy and moduli space of Yang-0Mills monopoles}}\label{complexrational_cohomotopy_and_moduli_space_of_yang0mills_monopoles}

The assignment of [[scattering amplitudes]] of [[monopoles]] in [[SU(2)]]-[[Yang-Mills theory]] is a [[diffeomorphism]]

\begin{displaymath}
\mathcal{M}_k
  \overset{ S }{\longrightarrow}
  R_k
\end{displaymath}
identifying the [[moduli space of monopoles]] of number $k$ with the space of complex-[[rational functions]] form the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k$ (hence the [[cocycle space]] of complex-rational 2-[[Cohomotopy]]).

(\href{moduli+space+of+monopoles#AtiyahHitchin88}{Atiyah-Hitchin 88, Theorem 2.10})

 $\,$

This is a [[non-abelian group|non-abelian]] analog of the [[Dirac charge quantization]] of the [[electromagnetic field]], with [[ordinary cohomology]] replaced by [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]].

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

\hypertarget{On4Manifolds}{}\subsubsection*{{Of 4-Manifolds}}\label{On4Manifolds}

Let $X$ be a [[4-manifold]] which is [[connected topological space|connected]] and [[orientation|oriented]].

The [[Pontryagin-Thom construction]] as \hyperlink{RelationToCobordismGroup}{above} gives for $n \in \mathbb{Z}$ the [[commuting diagram]] of sets

\begin{displaymath}
\itexarray{
     \pi^n(X) 
       &\overset{\simeq}{\longrightarrow}&
     \mathbb{F}_{4-n}(X)
     \\
     {}^{ \mathllap{h^n} }
     \downarrow
       &&
     \downarrow^{ h_{4-n} }
     \\
     H^n(X,\mathbb{Z})
       &\underset{\simeq}{\longrightarrow}&
     H_{4-n}(X,\mathbb{Z})
     \,,
  }
\end{displaymath}
where $\pi^\bullet$ denotes cohomotopy sets, $H^\bullet$ denotes [[ordinary cohomology]], $H_\bullet$ denotes [[ordinary homology]] and $\mathbb{F}_\bullet$ is [[normal bundle|normally]] [[framing|framed]] [[cobordism classes]] of [[normal bundle|normally]] [[framing|framed]] [[submanifolds]]. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of [[Eilenberg-MacLane spaces]]):

\begin{displaymath}
h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z}
  \,.
\end{displaymath}
Now

\begin{itemize}%
\item $h^0$, $h^1$, $h^4$ are [[isomorphisms]]


\item $h^3$ is an isomorphism if $X$ is ``odd'' in that it contains at least one closed oriented [[surface]] of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-[[quotient group]] of $\pi^3(X)$ (which is a group via the [[group]]-[[structure]] of the [[3-sphere]] ([[special unitary group|SU(2)]]))



\end{itemize}
(\hyperlink{KirbyMelvinTeichner12}{Kirby-Melvin-Teichner 12})

$\backslash$linebreak

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

[[!include flavours of cohomotopy -- table]]

$\backslash$linebreak

[[!include Segal completion -- table]]

$\backslash$linebreak

\begin{itemize}%
\item [[Cohomotopy charge map]]

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

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

Original articles include

\begin{itemize}%
\item [[K. Borsuk]], \emph{Sur les groupes des classes de transformations continues}, CR Acad. Sci. Paris 202.1400-1403 (1936): 2 (\href{https://doi.org/10.24033/asens.603}{doi:10.24033/asens.603})


\item [[Edwin Spanier]], \emph{Borsuk's Cohomotopy Groups}, Annals of Mathematics Second Series, Vol. 50, No. 1 (Jan., 1949), pp. 203-245 (\href{http://www.jstor.org/stable/1969362}{jstor:1969362})


\item Franklin P. Peterson, \emph{Some Results on Cohomotopy Groups}, American Journal of Mathematics Vol. 78, No. 2 (Apr., 1956), pp. 243-258 (\href{https://www.jstor.org/stable/2372514}{jstor:2372514})


\item [[Jan Jaworowski]], \emph{Generalized cohomotopy groups as limit groups}, Fundamenta Mathematicae 50 (1962), 393-402 (\href{https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/4}{doi:10.4064/fm-50-4-333-340}, \href{http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50133.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Cohomotopy_group}{Cohomotopy group}}


\item [[eom]], \emph{\href{https://www.encyclopediaofmath.org/index.php/Cohomotopy_group}{Cohomotopy group}}



\end{itemize}
The unstable [[Pontrjagin-Thom theorem]] identifying [[cobordism classes]] of [[normally framed submanifolds]] with their [[Cohomotopy charge]] is discussed for instance 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}
Further discussion includes

\begin{itemize}%
\item [[Laurence Taylor]], \emph{The principal fibration sequence and the second cohomotopy set}, Proceedings of the Freedman Fest, 235251, Geom. Topol. Monogr., 18, Coventry, 2012 (\href{https://arxiv.org/abs/0910.1781}{arXiv:0910.1781})


\item [[Robion Kirby]], [[Paul Melvin]], [[Peter Teichner]], \emph{Cohomotopy sets of 4-manifolds}, Geometry \& Topology Monographs 18 (2012) 161–190 (\href{https://arxiv.org/abs/1203.1608}{arXiv:1203.1608})


\item [[Martin Čadek]], Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, \emph{Computing all maps into a sphere}, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (\href{https://arxiv.org/abs/1105.6257}{arxiv:1105.6257})



\end{itemize}
\hypertarget{cohomotopy_cocycle_spaces}{}\subsubsection*{{Cohomotopy cocycle spaces}}\label{cohomotopy_cocycle_spaces}

Discussion of Cohomotopy [[cocycle spaces]] (i.e. [[spaces of maps]] into an [[n-sphere]]):

\begin{itemize}%
\item [[Vagn Lundsgaard Hansen]], \emph{The homotopy problem for the components in the space of maps on the $n$-sphere}, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (\href{https://doi.org/10.1093/qmath/25.1.313}{DOI:10.1093/qmath/25.1.313})


\item [[Vagn Lundsgaard Hansen]], \emph{On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere}, Transactions of the American Mathematical Society Vol. 265, No. 1 (May, 1981), pp. 273-281 (\href{https://www.jstor.org/stable/1998494}{jstor:1998494})



\end{itemize}
Discussion of [[cocycle spaces]] for [[rational Cohomotopy]] (see also at \emph{[[rational model of mapping spaces]]}):

\begin{itemize}%
\item [[Jesper Møller]], [[Martin Raussen]], \emph{Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces}, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (\href{https://www.jstor.org/stable/2000242}{jstor:2000242})

\end{itemize}
\hypertarget{relation_to_knots_and_links}{}\subsubsection*{{Relation to knots and links}}\label{relation_to_knots_and_links}

Relation to [[knots]] and [[links]]:

\begin{itemize}%
\item Maths.SE, \emph{\href{https://math.stackexchange.com/q/426482/58526}{Framed Cobordism Classes of links in $\mathbb{R}^3$}}

\end{itemize}
\hypertarget{application_to_persistency}{}\subsubsection*{{Application to persistency}}\label{application_to_persistency}

Application of Cohomotopy similar to that of [[persistent homology]]:

\begin{itemize}%
\item [[Peter Franek]], [[Marek Krčál]], \emph{Persistence of Zero Sets}, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (\href{https://arxiv.org/abs/1507.04310}{arXiv:1507.04310}, \href{http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16}{doi:10.4310/HHA.2017.v19.n2.a16})


\item [[Peter Franek]], [[Marek Krčál]], \emph{Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory}, talk at \href{https://www2.ist.ac.at/acat}{ACAT meeting 2015} (\href{https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf}{pfd slides})


\item [[Peter Franek]], [[Marek Krčál]], Hubert Wagner, \emph{Solving equations and optimization problems with uncertainty}, J Appl. and Comput. Topology (2018) 1: 297 (\href{https://arxiv.org/abs/1607.06344}{arxiv:1607.06344}, \href{https://doi.org/10.1007/s41468-017-0009-6}{doi:10.1007/s41468-017-0009-6})



\end{itemize}
\hypertarget{equivariant_cohomotopy}{}\subsubsection*{{Equivariant Cohomotopy}}\label{equivariant_cohomotopy}

Discussion of the [[stable cohomotopy|stable]] cohomotopy ([[framed manifold|framed]] [[cobordism cohomology theory]]) in the [[equivariant cohomology]]-version of cohomotopy ([[equivariant cohomotopy]]):

\begin{itemize}%
\item [[Arthur Wasserman]], section 3 of \emph{Equivariant differential topology}, Topology Vol. 8, pp. 127-150, 1969 (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf}{pdf})


\item [[James Cruickshank]], \emph{Twisted homotopy theory and the geometric equivariant 1-stem}, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 ()



\end{itemize}
and in the [[twisted cohomology]]-version ([[twisted cohomotopy]])

\begin{itemize}%
\item [[James Cruickshank]], Section 7 of \emph{Twisted homotopy theory and the geometric equivariant 1-stem}, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 ()

\end{itemize}
Discussion of [[M-brane]] physics in terms of [[equivariant rational homotopy theory|rational equivariant]] cohomotopy:

\begin{itemize}%
\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]} (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987})

\end{itemize}
and in terms of [[twisted cohomotopy]]:

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of \emph{[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]} (\href{https://arxiv.org/abs/1904.10207}{arXiv:1904.10207})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]} (\href{https://arxiv.org/abs/1906.07417}{arXiv:1906.07417})


\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}
\hypertarget{in_spectral_geometry}{}\subsubsection*{{In spectral geometry}}\label{in_spectral_geometry}

Discussion of [[smooth functions]] into the [[4-sphere]] in the context of [[Connes-Lott models]] in spectral [[non-commutative geometry]]:

\begin{itemize}%
\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Quanta of Geometry: Noncommutative Aspects}, Phys. Rev. Lett. 114 (2015) 9, 091302 (\href{https://arxiv.org/abs/1409.2471}{arXiv:1409.2471})


\item [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, \emph{Geometry and the Quantum: Basics}, JHEP 12 (2014) 098 (\href{https://arxiv.org/abs/1411.0977}{arXiv:1411.0977})


\item [[Alain Connes]], section 4 of \emph{Geometry and the Quantum}, in \emph{Foundations of Mathematics and Physics One Century After Hilbert}, Springer 2018. 159-196 (\href{https://arxiv.org/abs/1703.02470}{arXiv:1703.02470}, \href{https://www.springer.com/gp/book/9783319648125}{doi:10.1007/978-3-319-64813-2})



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

[[!redirects cohomotopy theory]] [[!redirects Cohomotopy theory]]

[[!redirects cohomotopy set]] [[!redirects cohomotopy sets]] [[!redirects Cohomotopy set]] [[!redirects Cohomotopy sets]]

[[!redirects cohomotopy group]] [[!redirects cohomotopy groups]] [[!redirects Cohomotopy group]] [[!redirects Cohomotopy groups]]

[[!redirects cohomotopy cohomology ]] [[!redirects cohomotopy cohomology theory]] [[!redirects Cohomotopy cohomology ]] [[!redirects Cohomotopy cohomology theory]]



\end{document}