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\begin{document}

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\section*{twisted 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{TwistedPontrjaginThomTheorem}{Twisted Pontrjagin-Thom theorem}\dotfill \pageref*{TwistedPontrjaginThomTheorem} \linebreak
\noindent\hyperlink{poincarhopf_theorem}{Poincaré–Hopf theorem}\dotfill \pageref*{poincarhopf_theorem} \linebreak
\noindent\hyperlink{equivariant_hopf_degree_theorem}{Equivariant Hopf degree theorem}\dotfill \pageref*{equivariant_hopf_degree_theorem} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Twisted Cohomotopy} is the [[twisted cohomology]]-variant of the the [[non-abelian cohomology]]-[[generalized cohomology theory|theory]] \emph{[[Cohomotopy]]}, [[representable functor|represented]] by [[homotopy types]] of [[n-spheres]].

The [[coefficients]]/[[twisted cohomology|twist]] for twisted Cohomotopy are [[spherical fibrations]], and [[cocycles]] are [[sections]] of these. For those [[spherical fibrations]] arising as [[unit sphere bundles]] of [[real vector bundles]] the twist may be understood as given by the [[J-homomorphism]].

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

Various classical theorems of [[differential topology]] are secretly theorems about [[twisted cohomotopy]]

\begin{quote}%
table grabbed from \hyperlink{FSS19b}{FSS 19b}


\end{quote}
\hypertarget{TwistedPontrjaginThomTheorem}{}\subsubsection*{{Twisted Pontrjagin-Thom theorem}}\label{TwistedPontrjaginThomTheorem}

The [[scanning map]]-equivalences on [[configuration spaces of points]] may be regarded as generalizations of the \href{cohomotopy#RelationToCobordismGroup}{Pontryagin-Thom theorem} from [[sets]] of [[Cohomotopy]] [[cohomology class|classes]] to [[homotopy types]] of [[twisted Cohomotopy]] [[cocycles]]:

\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 [[scanning 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{scanning 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 [[scanning 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{scanning 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{poincarhopf_theorem}{}\subsubsection*{{Poincaré–Hopf theorem}}\label{poincarhopf_theorem}

See at

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

\end{itemize}
\hypertarget{equivariant_hopf_degree_theorem}{}\subsubsection*{{Equivariant Hopf degree theorem}}\label{equivariant_hopf_degree_theorem}

On [[flat orbifolds]], twisted Cohomotopy becomes [[equivariant Cohomotopy]] and the twisted [[Hopf degree theorem]] becomes the

\begin{itemize}%
\item [[equivariant Hopf degree theorem]]

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

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

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

The concept is implicit in classical texts on [[differential topology]], for instance

\begin{itemize}%
\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]], Section 2 of: \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}
Discussion for twisted [[stable cohomotopy]] ([[framed manifold|framed]] [[cobordism cohomology theory]]):

\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 unstabilized twisted cohomotopy, with application to foundations of [[M-theory]]:

\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 twisted Cohomotopy]]

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



\end{document}