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\begin{document}

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\section*{equivariant 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{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - 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{equivariant_hopf_degree_theorem}{Equivariant Hopf degree theorem}\dotfill \pageref*{equivariant_hopf_degree_theorem} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{equivariant_cohomotopy_of__in_rodegree_}{Equivariant Cohomotopy of $S^V$ in RO-degree $V$}\dotfill \pageref*{equivariant_cohomotopy_of__in_rodegree_} \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{for_mbrane_charge_quantization}{For M-brane charge quantization}\dotfill \pageref*{for_mbrane_charge_quantization} \linebreak


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

The [[Bredon cohomology]]-[[equivariant cohomology|equivariant]] enhancement of [[cohomotopy]]-theory is \emph{equivariant cohomotopy}:

For $G$ a [[group]], $V$ a [[finite dimensional vector space|finite-dimensional]] [[real vector space|real]] $G$-[[representation]] $G \to O(V)$ and writing $S^V$ for the corresponding [[representation sphere]], the \emph{equivariant cohomotopy} in [[RO(G)-degree]] $V$ of a [[G-space]] $X$ is the set of $G$-[[equivariant homotopy theory|equivariant]] [[homotopy classes]] of maps from $X$ to $S^V$:

\begin{displaymath}
\pi_G^V\big( X \big)
  \;\coloneqq\;
  \left[
    X, S^V
  \right]^G
  \,.
\end{displaymath}
For $V = \mathbb{R}^n$ the [[trivial representation]] of [[dimension]] $n$, this reduces to the definition of plain [[cohomotopy]]-sets

\begin{displaymath}
\pi^{\mathbb{R}^n}(X)
  \;=\;
  \pi^n(X)
  \;=\;
  \left[ X, S^n\right]
  \,.
\end{displaymath}
The [[stabilization]] of this construction, in the sense of [[equivariant stable homotopy theory]], yields the concept of \emph{[[equivariant stable cohomotopy]]}.

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

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

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

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

\hypertarget{equivariant_cohomotopy_of__in_rodegree_}{}\subsubsection*{{Equivariant Cohomotopy of $S^V$ in RO-degree $V$}}\label{equivariant_cohomotopy_of__in_rodegree_}

As a special case of the [[equivariant Hopf degree theorem]] , we obtain the following:

\begin{example}
\label{EquivariantHomotopyOfSVInRODegreeV}\hypertarget{EquivariantHomotopyOfSVInRODegreeV}{}
\textbf{([[equivariant cohomotopy]] of [[representation sphere]] $S^V$ in [[RO(G)-degree]] $V$)}

Let $G \in \mathrm{Grp}_{\mathrm{fin}}$ and $V \in \mathrm{RO}(G)$ with $V^G = 0$. Then the bipointed [[equivariant cohomotopy]] of the [[representation sphere]] $S^V$ in [[RO(G)-degree]] $V$ is the [[Cartesian product]] of one copy of the [[integers]] for each [[isotropy group|isotropy]] subgroup \eqref{IsotropySubgroups} of $G$ in $S^V$ except the full subgroup $G \subset G$

\begin{displaymath}
\itexarray{
      \pi^V\left( S^V\right)^{\{0,\infty\}/}
      &
        \overset{\simeq}{\longrightarrow}
      &
      \underset{
        {
          {
              {H \in \mathrm{Isotr}_{S^V}(G)}
              \atop
              {H \neq G}
          }
        }
      }{\prod}
      \;\;
      {\vert W_G(H)\vert } \cdot \mathbb{Z}
      \\
      \big[
        S^V
          \overset{c}{\longrightarrow}
        S^V
      \big]
      &\mapsto&
      \Big(
        H 
          \mapsto
        \mathrm{deg}
        \big(
          c^H
        \big)
        -
        \mathrm{offs}(c,H)
      \Big)
  }
\end{displaymath}
where on the right

\begin{displaymath}
\mathrm{deg}
    \Big(
      \big(
        S^V
      \big)^H
      \overset{
        c^H
      }{\longrightarrow}
      \big(
        S^V
      \big)^H
    \Big)
    \in \mathbb{Z}
\end{displaymath}
is the [[integer]] [[winding number]] of the underlying [[continuous function]] of $c$ ([[corestriction|co]])[[restriction|restricted]] to $H$-[[fixed points]], and part of the claim is that this is an integer multiple of the order of the [[Weyl group]] $W_G(H)$ up to an offset

\begin{displaymath}
\mathrm{offs}(f,H) 
  \;\in\; 
  \big\{
    0,1, \cdots,  \left\vert W_G(H)\right\vert
  \big\}
  \;\subset\;
  \mathbb{Z}
\end{displaymath}
which depends in a definite way on the degrees of $c^K$ for all isotropy groups $K \gt H$.

\end{example}
For \textbf{[[proof]]} see \href{Hopf+degree+theorem#EquivariantHomotopyOfSVInRODegreeV}{here} at \emph{[[equivariant Hopf degree theorem]]}.

\begin{example}
\label{EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}\hypertarget{EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}{}
\textbf{([[equivariant cohomotopy]] of $S^{\mathbb{R}_{sgn}}$ in [[RO(G)-degree]] the [[sign representation]] $\mathbb{R}_{sgn}$)}

Let $G = \mathbb{Z}_2$ the [[cyclic group of order 2]] and $\mathbb{R}_{sgn} \in RO(\mathbb{Z}_2)$ its 1-dimensional [[sign representation]].

Under equivariant [[stereographic projection]] (\href{representation\+sphere#Construction}{here}) the corresponding [[representation sphere]] $S^{\mathbb{R}_{sgn}}$ is equivalently the [[unit circle]]

\begin{displaymath}
S^1 
  \simeq 
  S(\mathbb{R}^2)
\end{displaymath}
equipped with the $\mathbb{Z}_2$-[[action]] whose [[involution]] element $\sigma$ [[reflection|reflects]] one of the two [[coordinate functions|coordinates]] of the ambient [[Cartesian space]]

\begin{displaymath}
\sigma \;\colon\; (x_1,x_2) \mapsto (x_1, -x_2)
  \,.
\end{displaymath}
Equivalently, if we identify

\begin{equation}
S^1
  \;\simeq\;
  \mathbb{R}/\mathbb{Z}
\label{CircleAsQuaotientOfRByZ}\end{equation}
then the involution action is

\begin{displaymath}
\begin{aligned}
    \sigma
    \;\colon\;
    t 
    \mapsto
    &
    \phantom{\sim}
    1 - t 
    \\
    & \sim
    \phantom{1} - t
  \end{aligned} 
  \,.
\end{displaymath}
This means that the [[fixed point space]] is the [[0-sphere]]

\begin{displaymath}
\big( S^1\big)^{\mathbb{Z}_2}
  \;\simeq\;
  S^0
\end{displaymath}
being two antipodal points on the circle, which in the presentation \eqref{CircleAsQuaotientOfRByZ} are labeled $\{0,1/2\} \simeq S^0$.

Notice that the map

\begin{equation}
\itexarray{
    S^1 &\overset{n}{\longrightarrow}& S^1
    \\
    t &\mapsto& n\cdot t
  }
\label{ConstantParameterFunctionFromSignRepresentationSphereToItself}\end{equation}
of constant parameter speed and [[winding number]] $n \in \mathbb{N}$ is equivariant for this $\mathbb{Z}_2$-[[action]] on both sides:

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} t $\backslash$ar@\{|-{\tt \symbol{62}}\}r $\backslash$ar@\{|-{\tt \symbol{62}}\}d\emph{\{$\backslash$sigma\} \& n$\backslash$cdot t $\backslash$ar@\{|-{\tt \symbol{62}}\}d{\tt \symbol{94}}\{$\backslash$sigma\} $\backslash$ -t $\backslash$ar@\{|-{\tt \symbol{62}}\}r \& -n $\backslash$cdot t $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

Now the restriction of the map $n \cdot (-)\in \mathbb{Z}$ from \eqref{ConstantParameterFunctionFromSignRepresentationSphereToItself} to the [[fixed points]]

\begin{displaymath}
\itexarray{
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right) 
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
    \\
    {}^{ \mathllap{  \left( \cdot n\right)^{\mathbb{Z}_2} } }
    \big\downarrow 
      && 
    \big\downarrow^{\mathrlap{\cdot n}}
    \\
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right) 
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
  }
\end{displaymath}
sends (0 to 0 and) $1/2$ to either $1/2$ or to $0$, depending on whether the [[winding number]] is [[odd number|odd]] or [[even number|even]]:

\begin{displaymath}
\itexarray{
     S^0 
       &\overset{ \left(\cdot n\right)^{\mathbb{Z}_2} }{\longrightarrow}&
     S^0
     \\
     1/2 &\mapsto&
     \left\{
       \itexarray{
         1/2 &\vert& n \;\text{is odd}
         \\
         0 &\vert& n \text{is even}
       }
     \right.
  }
\end{displaymath}
Hence if the restriction to the [[fixed locus]] is taken to be the [[identity function|identity]] (bipointed [[equivariant cohomotopy]]) then, in accord with Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} there remains the [[integers]] worth of equivariant [[homotopy classes]], where each integer $k \in \mathbb{Z}$ corresponds to the odd winding integer $1 + 2k$

\begin{displaymath}
\itexarray{
    \pi^{\mathbb{R}_{sgn}}
    \left(
      S^{\mathbb{R}_{sgn}}
    \right)^{\{0,\infty\}/}
    &\simeq&
    2 \cdot \mathbb{Z} + 1 
    &\simeq& 
   \mathbb{Z}
    \\
    \left[
      \mathbb{R}/\mathbb{Z} \overset{c}{\to} \mathbb{R}/\mathbb{Z}
    \right]_{{0 \mapsto 0} \atop {1/2 \mapsto 1/2}}
    &\mapsto&
    deg(c)
    &\mapsto&
    \big( deg(c) - 1\big)/2
  }
\end{displaymath}
\end{example}
\begin{example}
\label{EquivariantCohomotopyOfRepresentationSphereOfQuaternionsInThatDegree}\hypertarget{EquivariantCohomotopyOfRepresentationSphereOfQuaternionsInThatDegree}{}
\textbf{([[equivariant cohomotopy]] of $S^{\mathbb{H}}$ in [[RO(G)-degree]] the [[quaternions]] $\mathbb{H}$)}

Let $G \subset SU(2) \simeq S(\mathbb{H})$ be a non-[[trivial group|trivial]] [[finite subgroup of SU(2)]] and let $\mathbb{H} \in RO(G)$ be the [[real vector space]] of [[quaternions]] regarded as a [[linear representation]] of $G$ by left multiplication with unit [[quaternions]].

Then the bi-pointed [[equivariant cohomotopy]] of the [[representation sphere]] $S^{\mathbb{H}}$ in [[RO(G)-degree]] $\mathbb{H}$ is

\begin{displaymath}
\itexarray{
    \pi^{\mathbb{H}}
    \left(
      S^{\mathbb{H}} 
    \right)^{\{0,\infty\}/}
    &\simeq&
    {\left\vert G\right\vert} \cdot \mathbb{Z} + 1
    &\simeq& 
    {\left\vert G\right\vert} \cdot \mathbb{Z}
    &\simeq& 
    \mathbb{Z}    
    \\
    \left[   
      S^{\mathbb{H}}
      \overset{c}{\longrightarrow}
       S^{\mathbb{H}}
    \right]
    &\mapsto&
    deg\left( c^{ \{e\}  }\right)
    &\mapsto&
    deg\left( c^{ \{e\}  }\right) - 1
    &\mapsto&
    \big( 
      deg\left( c^{ \{e\}  }\right) - 1
    \big)/ {\left\vert G\right\vert}
  }
\end{displaymath}
\end{example}
\begin{proof}
The only [[isotropy groups|isotropy]] [[subgroups]] of the left action of $G$ on $\mathbb{H}$ are the two extreme cases $Isotr_{\mathbb{H}}(G) = \{1, G\} \in Sub(G)$. Hence the only multiplicity that appears in Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} is

\begin{displaymath}
\left\vert W_G(1)\right\vert 
  \;=\;
  \left\vert G \right\vert   
  \,.
\end{displaymath}
and all degrees must differ from that of the class of the [[identity function]] by a multiple of this multiplicity. Finally, the offset of the identity function is clearly $offs\left(id_{S^{\mathbb{H}}},1\right) = deg\left( id_{S^{\mathbb{H}}}\right) = 1$.

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

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

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

\end{itemize}


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

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

\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 [[Tammo tom Dieck]], section 8.4 of \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766 Springer 1979


\item [[George Peschke]], \emph{Degree of certain equivariant maps into a representation sphere}, Topology and its Applications Volume 59, Issue 2, 30 September 1994, Pages 137-156 ()


\item Zalman Balanov, \emph{Equivariant hopf theorem}, Nonlinear Analysis: Theory, Methods \& Applications Volume 30, Issue 6, December 1997, Pages 3463-3474 ()


\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 ()


\item Davide L. Ferrario, \emph{On the equivariant Hopf theorem}, \emph{Topology Volume 42, Issue 2, March 2003, Pages 447-465} ()



\end{itemize}
\hypertarget{for_mbrane_charge_quantization}{}\subsubsection*{{For M-brane charge quantization}}\label{for_mbrane_charge_quantization}

Discussion of [[M-brane]] [[charge quantization]] in [[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})

([[equivariant rational homotopy theory|equivariant]] [[rational Cohomotopy]])


\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})

(implying [[RR-field tadpole cancellation]])



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

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



\end{document}