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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Hopf degree theorem} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{algebraic_topology}{}\paragraph*{{Algebraic topology}}\label{algebraic_topology} [[!include algebraic topology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak \noindent\hyperlink{in_ordinary_homotopy_theory}{In ordinary homotopy theory}\dotfill \pageref*{in_ordinary_homotopy_theory} \linebreak \noindent\hyperlink{InEquivariantHomotopyTheory}{In equivariant homotopy theory}\dotfill \pageref*{InEquivariantHomotopyTheory} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \noindent\hyperlink{related_statements}{Related statements}\dotfill \pageref*{related_statements} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{statement}{}\subsection*{{Statement}}\label{statement} \hypertarget{in_ordinary_homotopy_theory}{}\subsubsection*{{In ordinary homotopy theory}}\label{in_ordinary_homotopy_theory} \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]] with 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)}, \hyperlink{Kobin16}{Kobin 16, 7.5}) \hypertarget{InEquivariantHomotopyTheory}{}\subsubsection*{{In equivariant homotopy theory}}\label{InEquivariantHomotopyTheory} The [[equivariant Hopf degree theorem]] -- Theorem \ref{EquivariantHopfDegreeTheorem} below -- is a generalization of the Hopf degree theorem to [[equivariant homotopy theory]], due to \hyperlink{tomDieck79}{tomDieck 79, 8.4}. It implies a fairly explicit characterization of [[equivariant cohomotopy]] of [[representation spheres]] $S^V$ in [[RO(G)-degree]] $V$ (Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} below). We need the following list of ingredients and assumptions: Let $G$ be a [[finite group]]. For $H \subset G$ a [[subgroup]], write \begin{equation} W_G H \coloneqq (N_G H) / H \label{WeylGroup}\end{equation} for its \href{Weyl+group#in_equivariant_homotopy_theory}{Weyl group}. For $X$ a [[G-space]], we write \begin{equation} Isotr_X(G) \subset Sub(G) \label{IsotropySubgroups}\end{equation} for the [[subset]] of the [[subgroup lattice]] on the [[isotropy groups]] of $X$, hence those [[subgroups]] which appear as [[stabilizer subgroups]] $Stab_G(x)$ of some point $x \in X$. This means that if $H_1, H_2 \in Iso_X(G)$ and $H_1 \lt H_2$ is a strict inclusion, then the [[fixed loci]] differ $X^{H_1} \gt X^{H_2}$. \begin{defn} \label{MatchingGSpaces}\hypertarget{MatchingGSpaces}{} \textbf{(matching pair of $G$-spaces)} For $G$ a [[finite group]], we say that a [[pair]] $X,Y \in G Spaces$ of [[topological G-spaces]], where $X$ is a [[G-CW-complex]], is a \emph{matching pair} if the following conditions are satisfied for all [[isotropy groups]] $H \in Isotr_X(G)$ \eqref{IsotropySubgroups} of the $G$-action on $X$. \begin{enumerate}% \item The [[fixed point space]] $X^H$ is a $W_G H = (N_G H) / H$[[G-CW-complex|-complex]] of [[finite number|finite]] [[dimension of a cell complex|dimension]] $dim\big( X^H\big) \in \mathbb{N}$; \item $H^{dim(X^H)}\Big( X^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ ([[integral cohomology]] of the [[fixed point space]]), this implies that the [[action]] of $W_G(H)$ on cohomology induces a [[group homomorphism]] \begin{equation} e_{H,X} \;\colon\; W_G(H) \longrightarrow Aut_{Ab}(\mathbb{Z}) \simeq \mathbb{Z}^\times \label{OrientationBehaviour}\end{equation} to be called the \emph{orientation behaviour} of the action of $W_G(H)$ on $X^H$; \end{enumerate} and \begin{enumerate}% \item $Y^H$ is $(dim(X^H)-1)$-[[n-connected topological space|connected]] \begin{displaymath} pi_{ \lt \mathrm{dim}\big( X^H \big) } \big( Y^H \big) \;=\; 0 \end{displaymath} (hence [[connected topological space|connected]] if $dim\left(X^H\right) = 1$, [[simply connected topological space|simply connected]] if $dim\left(X^H\right) = 2$, etc.); \item $\pi_{dim(X^H)}\big( Y^H\big) \simeq \mathbb{Z}$ ([[homotopy groups]] of [[fixed point space]]), with the previous point this implies (by the [[Hurewicz theorem]]) that $H^{dim(X^H)}\Big( Y^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ and hence orientation behaviour \eqref{OrientationBehaviour} $e_{H,Y} \;\colon\; W_G(H) \to \mathbb{Z}^\times$ \item $e_{H,X} = e_{H,Y}$, the orientation behaviour \eqref{OrientationBehaviour} of $X$ and $Y$ agrees at all [[isotropy groups]]. \end{enumerate} For simplicity we also demand that \begin{itemize}% \item $dim(X^H) \geq 1$. \end{itemize} Given a matching pair of $G$-spaces, we say that a choice of generators in the [[cohomology groups]] of all the fixed strata \begin{equation} \begin{aligned} or_{X,H} & = \pm 1 \in \mathbb{Z} \simeq H^{dim(X^H)}\big(X^H, \mathbb{Z} \big) \\ or_{Y,H} & = \pm 1 \in \mathbb{Z} \simeq H^{dim(X^H)}\big(Y^H, \mathbb{Z} \big) \end{aligned} \label{ChoiceOfStratumWiseOrientations}\end{equation} is a \emph{choice of singularity-wise [[orientations]]}. Given such a choice and an equivariant continuous function $f \colon X \to Y$, we have for each [[isotropy group]] $H \in Isotr_X(G)$ that the continuous function $f^H \;\colon\; X^H \to Y^H$ has a well-defined [[integer]] [[degree of a continuous function|degree]] \begin{equation} deg(f^H) \; \in \;\mathbb{Z} \,. \label{IntegerDegreeOfEquivariantFunctionOnFixedStrata}\end{equation} \end{defn} (\hyperlink{tomDieck79}{tom Dieck 79, p. 212 and p. 213}) \begin{example} \label{RepresentationSphereToRepresentationSphereIsMatchingPair}\hypertarget{RepresentationSphereToRepresentationSphereIsMatchingPair}{} Let $G$ be a [[finite group]] and $V \in RO(G)$ a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] [[linear representation]] of $G$. Then the pair \begin{displaymath} (X \coloneqq S^V,\; Y \coloneqq S^V) \end{displaymath} consisting of two copies of the [[representation sphere]] of $V$ is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}. \end{example} \begin{proof} First notice that an $H$-[[fixed locus]] of any $G$-representation sphere is itself a $W_G(H)$-representation sphere, since $\left(S^V\right)^H \simeq S^{\left( V^H\right)}$. That these are all [[G-CW-complex|WH-CW-complexes]] follows because generally [[G-representation spheres are G-CW-complexes]]. Moreover, since the [[topological spaces]] underlying all these fixed loci are [[n-spheres]], and since we have the same spheres for $X$ and $Y$, the connectivity and orientability conditions in Def. \ref{MatchingGSpaces} are evidently satisfied. \end{proof} \begin{example} \label{RepresentationTorusToRepresentationSphereIsMatchingPair}\hypertarget{RepresentationTorusToRepresentationSphereIsMatchingPair}{} Let $G$ be a [[finite group]] which arises as the [[point group]] $G \simeq S/N$ of a [[crystallographic group]] $S \subset Iso(E)$ of some [[Euclidean space]] $E$. Then the pair \begin{displaymath} \big( X \coloneqq E/N \,; Y \coloneqq S^E \big) \end{displaymath} consisting of \begin{enumerate}% \item the [[torus]] $E/N$ which is the [[quotient space]] of $E$ by the given crystallographic sub-lattice $N \subset E$ and equipped with the $G$-[[action]] descending from that on $E$ (\href{crystallographic+group#InducedPointGroupActionOnTorus}{this Prop.}); \item the [[representation sphere]] of the [[linear representation|linear action]] of the point group $G$ on $E$ \end{enumerate} is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}. \end{example} \begin{proof} The [[torus]] $E/N$ carries the [[structure]] of a [[smooth manifold]] for which the $G$-[[action]] is [[smooth]]. Since $G$ is [[finite group|finite]], also all its [[fixed loci]] $(E/N)^H$ are smooth manifolds (\href{equivariant+differential+topology#FixedLociOfSmoothProperActionsAreSubmanifolds}{this Prop.}). By the [[equivariant triangulation theorem]], all these are [[G-CW-complexes|WH-CW-complexes]]. Moreover, the orientability and connectivity assumptions in Def. \ref{MatchingGSpaces} are evidently satisfied, using the fact that both $E/N$ as well as $S^E$ are modeled on the same linear $G$-representation space $E$. \end{proof} \begin{theorem} \label{EquivariantHopfDegreeTheorem}\hypertarget{EquivariantHopfDegreeTheorem}{} \textbf{([[equivariant Hopf degree theorem]])} Given a matching pair of $G$-spaces $X, Y$ (Def. \ref{MatchingGSpaces}) the function (from $G$-[[equivariant homotopy theory|equivariant]] [[homotopy classes]] to [[tuples]] of [[degree of a continuous function|degrees]] labeled by [[isotropy groups]]) which sends any equivariant homotopy class $[f]$ of an equivariant continuou sfunction $f \colon X \to Y$ to its $H$-degrees \eqref{IntegerDegreeOfEquivariantFunctionOnFixedStrata} \begin{displaymath} \itexarray{ deg &\colon& \pi_0 \mathrm{Maps}\big( X,Y \big)^G & \overset{ \phantom{AAAA} }{\hookrightarrow} & \underset{ { H \in Isotr_X(G) } }{\prod} \mathbb{Z} \\ && [f] &\mapsto & \big( H \mapsto \mathrm{deg}\big( f^H\big) \big) } \end{displaymath} is an [[injective function|injection]]. Moreover, for each $[f] \in \pi_0 \mathrm{Maps}\big( X,Y \big)^G$ and for each $H \in Isotropy_X(G)$ \begin{enumerate}% \item the $H$-degree \eqref{IntegerDegreeOfEquivariantFunctionOnFixedStrata} modulo the [[order of a group|order]] of the [[Weyl group]] \begin{equation} deg\left( f^H\right) \;mod\; {\vert W_G(H)\vert} \;\in\; \mathbb{Z}/{\vert W_G(H)\vert} \label{DegreeConstraintsInEquivariantHopfDegreeTheorem}\end{equation} is fixed by the degrees $deg\big( f^K\big)$ for all $K \gt H$; \item there exists $f'$ with specified degrees $deg\big( (f')^K\big) = deg\big( f^K\big)$ for $K \gt H$ and realizing any of the degrees $deg\big( (f')^H\big) \in \mathbb{Z}$ allowed by the constraint \eqref{DegreeConstraintsInEquivariantHopfDegreeTheorem}. \end{enumerate} \end{theorem} (\hyperlink{tomDieck79}{tom Dieck 79, 8.4}) As a special case of the [[equivariant Hopf degree theorem]] (Theorem \ref{EquivariantHopfDegreeTheorem}), we obtain the following: \begin{prop} \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 $V \in \mathrm{RO}(G)$. Then the pointed [[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 \emph{proper} [[isotropy group|isotropy]] subgroup \eqref{IsotropySubgroups} $H \underset{\neq}{\subset} G$ in $S^V$, and a copy of $\mathbb{Z}_2$ or $\mathbb{Z}$ depending on whether $V^G = 0$ or not: \begin{displaymath} \itexarray{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \itexarray{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \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 \left\{ \itexarray{ \mathbb{Z} &\vert& dim\left(V^H\right) \gt 0 \\ \mathbb{Z}_2 &\vert& dim\left( V^H\right) = 0 } \right\} \end{displaymath} is the [[winding number]] of the underlying [[continuous function]] of $c$ ([[corestriction|co]])[[restriction|restricted]] to $H$-[[fixed points]], and part of the claim is that in the cases with $dim\left( V^H\right) \gt 0$ this is an integer multiple of the order of the [[Weyl group]] $W_G(H)$ \eqref{WeylGroup} 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{prop} \begin{proof} This follows as a special case of the equivariant Hopf degree theorem (Theorem \ref{EquivariantHopfDegreeTheorem}). Here $(S^V, S^V)$ is a matching pair of $G$-spaces according to Example \ref{RepresentationSphereToRepresentationSphereIsMatchingPair}. This equivariant Hopf degree theorem is stated above under the simplifying assumption that the dimension of all fixed loci is positive. But the proof from \hyperlink{tomDieck79}{tomDieck 79, 8.4} immediately applies to our situation where the dimension of the fixed locus at the full subgroup $H = G$ may be 0, with $\left( S^V\right)^G = S^0$. This gives a choice in $\mathbb{Z}_2$ in the first step of the inductive argument in \hyperlink{tomDieck79}{tomDieck 79, 8.4}, and from there on the proof applies verbatim. Alternatively, if $V^G = 0$ we may consider maps $S^{1+V} \to S^{1+V}$ which restrict on $S^1 \to S^1$ to degree zero or one. \end{proof} \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{remark} \label{}\hypertarget{}{} This result \begin{displaymath} \pi^{ \mathbb{R}_{sgn} }\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z}_2 \times \mathbb{Z} \end{displaymath} from Example \ref{EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree} becomes, after [[stabilization]] to [[equivariant stable homotopy theory]], the stable homotopy groups of the [[equivariant sphere spectrum]] in [[RO(G)-grading]] given by \begin{displaymath} \pi^{ \mathbb{R}_{sgn} }_{stab}\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z} \times \mathbb{Z} \end{displaymath} see \href{equivariant+sphere+spectrum#Z2equivariance}{there}. \end{remark} \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{applications}{}\subsection*{{Applications}}\label{applications} \begin{itemize}% \item [[Burnside ring is equivariant stable cohomotopy of the point]] \end{itemize} \hypertarget{related_statements}{}\subsection*{{Related statements}}\label{related_statements} \begin{itemize}% \item [[Hurewicz theorem]] \item [[Boardman homomorphism]] \item [[Hopf invariant]] \item [[Poincaré–Hopf theorem]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Due to [[Heinz Hopf]]. Textbook accounts: \begin{itemize}% \item [[Raoul Bott]], [[Loring Tu]], Chapter 11 of \emph{[[Differential Forms in Algebraic Topology]]}, Graduate Texts in Mathematics 82, Springer 1982 (\href{https://doi.org/10.1007/BFb0063500}{doi:10.1007/BFb0063500}) \item B. A. Dubrovin, [[S. P. Novikov]], A. T. Fomenko, section 13.3 of: \emph{Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds}, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985 \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} Review: \begin{itemize}% \item Andrew Kobin, \emph{Algebraic Topology}, 2016 ([[KobinAT2016.pdf:file]]) \end{itemize} Generalization to [[equivariant cohomotopy]] and [[equivariant cohomology]] \begin{itemize}% \item [[Tammo tom Dieck]], section 8.4 of \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766 Springer 1979 \end{itemize} [[!redirects Hopf theorem]] \end{document}