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\begin{document}

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\section*{rational Cohomotopy}

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

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

[[!include cohomology - contents]]

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{sullivan_models_for_cocycle_spaces}{Sullivan models for cocycle spaces}\dotfill \pageref*{sullivan_models_for_cocycle_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_cocycle_space_for_}{The cocycle space for $\pi^4\big( S^1\big)_{\mathbb{Q}}$}\dotfill \pageref*{the_cocycle_space_for_} \linebreak
\noindent\hyperlink{the_cocycle_space_for__2}{The cocycle space for $\pi^4\big( S^3\big)_{\mathbb{Q}}$}\dotfill \pageref*{the_cocycle_space_for__2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{cocycle_spaces}{Cocycle spaces}\dotfill \pageref*{cocycle_spaces} \linebreak
\noindent\hyperlink{on_superspaces}{On super-spaces}\dotfill \pageref*{on_superspaces} \linebreak
\noindent\hyperlink{rational_equivariant_cohomotopy}{Rational equivariant Cohomotopy}\dotfill \pageref*{rational_equivariant_cohomotopy} \linebreak
\noindent\hyperlink{rational_twisted_cohomotopy}{Rational twisted Cohomotopy}\dotfill \pageref*{rational_twisted_cohomotopy} \linebreak


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

\emph{Rational Cohomotopy theory} $\pi^\bullet_{\mathbb{Q}}$ is the approximation in [[rational homotopy theory]] of [[Cohomotopy cohomology theory]], i.e. the ([[non-abelian cohomology|non-abelian]]) [[generalized cohomology theory]] whose [[cocycle spaces]] are [[rationalizations]] of the [[cocycle spaces]] of plain [[Cohomotopy cohomology theory]], hence which are [[spaces of maps]] into [[rational n-spheres]] $S^n_{\mathbb{Q}}$:

\begin{displaymath}
\pi^n_{\mathbb{Q}}(-)
  \;=\;
  \pi_0 Maps\big(-, S^n\big)_{\mathbb{Q}}
  \;\simeq\;
  \pi_0 Maps\big(-, S^n_{\mathbb{Q}}\big)
  \;.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{sullivan_models_for_cocycle_spaces}{}\subsubsection*{{Sullivan models for cocycle spaces}}\label{sullivan_models_for_cocycle_spaces}

Discussion of [[rational cohomology]] and of [[Sullivan models]] for [[cocycle spaces]] in rational Cohomotopy.

\begin{prop}
\label{RationalHomotopyTypeOfMapsNSphereToNsphere}\hypertarget{RationalHomotopyTypeOfMapsNSphereToNsphere}{}
\textbf{([[rational homotopy type]] of [[space of maps]] from [[n-sphere]] to itself)}

Let $n \in \mathbb{N}$ be a [[natural number]] and $f\colon S^n \to S^n$ a [[continuous function]] from the [[n-sphere]] to itself. Then the [[connected component]] $Maps_f\big( S^n, S^n\big)$ of the [[space of maps]] which contains this map has the following [[rational homotopy theory|rational]] [[homotopy type]]:

\begin{equation}
Maps_f\big( S^n, S^n\big)
  \;\simeq_{\mathbb{Q}}\;
  \left\{
    \itexarray{
      S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0
      \\
      S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0
      \\
      S^n &\vert& n\, \text{odd}
    }
  \right.
\label{RationalHomotopyTypeOfMappingSpacesSnToSn}\end{equation}
where $deg(f)$ is the [[degree of a continuous function|degree]] of $f$.

Moreover, under the canonical morphism expressing the canonical [[action]] of the [[special orthogonal group]] $SO(n+1)$ on $S^n = S\big( \mathbb{R}^{n+1}\big)$ (regarded as the [[unit sphere]] in $(n+1)$-[[dimension|dimensional]] [[Cartesian space]]) we have that on [[ordinary homology]]

\begin{displaymath}
\itexarray{
    H_\bullet\Big( 
      SO\big( n+ 1 \big)
    \Big)
    &\longrightarrow&
    H_\bullet\Big( 
      Maps_{f = id}\big( S^n, S^n  \big) 
    \Big)
  }
\end{displaymath}
the generator in $\left\{ \itexarray{ H_{2n+1}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{even} \\ H_{n}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{odd}   }  \right.$ maps to the [[fundamental class]] of the respective [[spheres]] in \eqref{RationalHomotopyTypeOfMappingSpacesSnToSn}, all other generators mapping to [[zero]].

\end{prop}
(\hyperlink{MollerRaussen85}{Møller-Raussen 85, Example 2.5}, \hyperlink{CohenVoronov05}{Cohen-Voronov 05, Lemma 5.3.5})

See at \emph{[[Sullivan model of a spherical fibration]]} for more on this.

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

Let

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

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

\begin{displaymath}
H^\bullet
  \big(
    \Omega^D S^n
    ,
    \mathbb{Q}
  \big)
  \;\simeq\;
  \mathbb{Q}\big[ \omega_{n - D}, \omega_{2n - 1 - D} \big] 
  \,.
\end{displaymath}
\end{prop}
(by \href{Sullivan+model+of+free+loop+space#SullivanModelsOfMapsFromSkToSnFornLargerk}{this Prop.} at \emph{[[Sullivan model of based loop space]]}; see also \hyperlink{KallelSjerve99}{Kallel-Sjerve 99, Prop. 4.10})

For the edge case $\Omega^D S^D$ the above formula does not apply, since $\Omega^{D-1} S^D$ is not [[simply connected topological space|simply connected]] (its [[fundamental group]] is $\pi_1\big( \Omega^{D-1}S^D \big) = \pi_0 \big(\Omega^D S^D\big) = \pi_D(S^D) = \mathbb{Z}$, the 0th [[stable homotopy group of spheres]]).

But:

\begin{example}
\label{RationalModelsForBasedMappingSpaceSDToSD}\hypertarget{RationalModelsForBasedMappingSpaceSDToSD}{}
The rational model for $\Omega^D S^D$ follows from Prop. \ref{RationalHomotopyTypeOfMapsNSphereToNsphere} by realizing the pointed mapping space as the [[homotopy fiber]] of the [[evaluation map]] from the free mapping space:

\begin{displaymath}
\itexarray{
    \mathllap{
      \Omega^D S^D
      \simeq
    \;}
    Maps^{\ast/\!}\big( S^D, S^D\big)
    \\
    \big\downarrow^{\mathrlap{fib(ev_\ast)}}
    \\
    Maps(S^D, S^D)
    \\
    \big\downarrow^{\mathrlap{ev_\ast}}
    \\
    S^D
  }
\end{displaymath}
This yields for instance the following examples.

In odd dimensions:

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}3, S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} $\backslash$ast $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}3, S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$ard{\tt \symbol{94}}-\{ ($\backslash$mathrm\{id\}\_\{S{\tt \symbol{94}}3\})\_\{n $\backslash$in $\backslash$mathbb\{N\}\} \} $\backslash$ S{\tt \symbol{94}}3 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}3 $\backslash$end\{xymatrix\}

In even dimensions:

(In the following $h_{\mathbb{K}}$ denotes the [[Hopf fibration]] of the [[division algebra]] $\mathbb{K}$, hence $h_{\mathbb{C}}$ denotes the [[complex Hopf fibration]] and $h_{\mathbb{H}}$ the [[quaternionic Hopf fibration]].)

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}2, S{\tt \symbol{94}}2 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}1 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}2, S{\tt \symbol{94}}2 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$big( S{\tt \symbol{94}}2 $\backslash$times S{\tt \symbol{94}}1 $\backslash$big) $\backslash$sqcup $\backslash$big( $\backslash$underset\{ n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$big( p\_1, (h\_\{$\backslash$mathbb\{C\}\})\_\{n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{N\}\} $\backslash$big) \} $\backslash$ S{\tt \symbol{94}}2 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}2 $\backslash$end\{xymatrix\}

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}4, S{\tt \symbol{94}}4 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}4, S{\tt \symbol{94}}4 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$big( S{\tt \symbol{94}}4 $\backslash$times S{\tt \symbol{94}}3 $\backslash$big) $\backslash$sqcup $\backslash$big( $\backslash$underset\{ n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}7 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$big( p\_1, (h\_\{$\backslash$mathbb\{H\}\})\_\{n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{N\}\} $\backslash$big) \} $\backslash$ S{\tt \symbol{94}}4 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}4 $\backslash$end\{xymatrix\}

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

We discuss examples of cocycle spaces for rational 4-Cohomotopy, hence with [[coefficients]] in the [[rational n-sphere|rational]] [[4-sphere]], whose [[Sullivan model]] is

\begin{displaymath}
CE
  \big( 
    S^4
  \big)
  \;=\;
  \left(
  \begin{aligned}
  d\,g_4 & = 0
  \\
  d\,g_7 & = -\tfrac{1}{2} g_4 \wedge g_4
  \end{aligned}
  \right)
  \,.
\end{displaymath}
\hypertarget{the_cocycle_space_for_}{}\subsubsection*{{The cocycle space for $\pi^4\big( S^1\big)_{\mathbb{Q}}$}}\label{the_cocycle_space_for_}

The [[Sullivan model]] for the [[space of maps]] $Maps(S^1, S^4)$ from the [[1-sphere]] to the [[4-sphere]] is

\begin{displaymath}
CE
  \Big(
    \mathfrak{l}
    Maps
    \big( 
      S^1, S^4
    \big)
  \Big)
  \;=\;
  \left(
  \begin{aligned}
    d\,h_3 & = 0
    \\
    d\, \omega_4 & = 0
    \\
    d\, \omega_6 & = h_3 \wedge \omega_4
    \\
    d\, h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  \end{aligned}
  \right)
\end{displaymath}
By \hyperlink{Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace}{this Prop.}, see \hyperlink{FSS16}{FSS 16, Section 3}.

\hypertarget{the_cocycle_space_for__2}{}\subsubsection*{{The cocycle space for $\pi^4\big( S^3\big)_{\mathbb{Q}}$}}\label{the_cocycle_space_for__2}

The [[Sullivan model]] for the [[space of maps]] $Maps(S^3, S^4)$ from the [[3-sphere]] to the [[4-sphere]] is

\begin{displaymath}
CE
  \Big( 
    \mathfrak{l}
    Maps
    \big(
      S^3, S^4
    \big)
  \Big)
  \;=\;
  \left(
  \begin{aligned}
    d\, b_1 & = 0
    \\
    d\, \omega_4 & = 0
    \\
    d\, v_{{}_{4}} & = \omega_4 \wedge b_1
    \\
    d\, \omega_7 & = - \tfrac{1}{2} \, \omega_4 \wedge \omega_4
  \end{aligned}
  \right)
\end{displaymath}
By \hyperlink{MollerRaussen85}{Møller-Raussen 85, Prop. 2.3}.

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

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

See also

\begin{itemize}%
\item [[rational cohomology]]

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

\hypertarget{cocycle_spaces}{}\subsubsection*{{Cocycle spaces}}\label{cocycle_spaces}

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

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


\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})


\item J.-B. Gatsinzi, \emph{Rational Gottlieb Group of Function Spacesof Maps into an Even Sphere}, International Journal of Algebra, Vol. 6, 2012, no. 9, 427 - 432 (\href{http://www.m-hikari.com/ija/ija-2012/ija-9-12-2012/gatsinziIJA9-12-2012.pdf}{pdf})



\end{itemize}
\hypertarget{on_superspaces}{}\subsubsection*{{On super-spaces}}\label{on_superspaces}

The observation that the [[equations of motion]] of the [[supergravity C-field]] and its dual in [[D=11 N=1 supergravity]] characterize it as a [[cocycle]] in rational 4-Cohomotopy:

\begin{itemize}%
\item [[Hisham Sati]], Section 2.5 of: \emph{Framed M-branes, corners, and topological invariants}, J. Math. Phys. 59 (2018), 062304 (\href{http://arxiv.org/abs/1310.1060}{arXiv:1310.1060})

\end{itemize}
Rational Cohomotopy of [[super-spaces]] (see also at \emph{[[geometry of physics -- fundamental super p-branes]]}):

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: \emph{[[schreiber:Rational sphere valued supercocycles in M-theory|Rational sphere valued supercocycles in M-theory and type IIA string theory]]}, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) (\href{https://arxiv.org/abs/1606.03206}{arXiv:1606.03206})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: \emph{[[schreiber:The WZW term of the M5-brane|The WZW term of the M5-brane and differential cohomotopy]]}, J. Math. Phys. 56, 102301 (2015) (\href{https://arxiv.org/abs/1506.07557}{arXiv:1506.07557})

(\textbf{\href{http://arxiv.org/abs/1506.07557}{arXiv:1506.07557}})



\end{itemize}
Review in:

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: \emph{[[schreiber:The rational higher structure of M-theory]]}, in: \emph{Proceedings of the \href{http://www.maths.dur.ac.uk/lms/}{LMS-EPSRC Durham Symposium}:} \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}, August 2018}m Fortschritte der Physik, 2019 (\href{https://arxiv.org/abs/1903.02834}{arXiv:1903.02834})

\end{itemize}
\hypertarget{rational_equivariant_cohomotopy}{}\subsubsection*{{Rational equivariant Cohomotopy}}\label{rational_equivariant_cohomotopy}

Discussion of [[rational equivariant 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]]}, Comm. Math. Phys. 2019 (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987})

\end{itemize}
\hypertarget{rational_twisted_cohomotopy}{}\subsubsection*{{Rational twisted Cohomotopy}}\label{rational_twisted_cohomotopy}

Discussion of rational [[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})



\end{itemize}
[[!redirects rational Cohomotopy theory]] [[!redirects rational Cohomotopy cohomology theory]]

[[!redirects rational cohomotopy]] [[!redirects rational cohomotopy theory]] [[!redirects rational cohomotopy cohomology theory]]



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