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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*{geometric fixed point spectrum} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory} [[!include stable homotopy theory - contents]] \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{ForEquivariantSuspensionSpectra}{For equivariant suspension spectra}\dotfill \pageref*{ForEquivariantSuspensionSpectra} \linebreak \noindent\hyperlink{InTermsOfSmashingLocalization}{In terms of smashing localization}\dotfill \pageref*{InTermsOfSmashingLocalization} \linebreak \noindent\hyperlink{AsInversionOfEulerClasses}{Via inversion of Euler classes}\dotfill \pageref*{AsInversionOfEulerClasses} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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} In the context of [[equivariant stable homotopy theory]] and $G$ one distinguishes, for a [[G-spectrum]] $E$, between the plain \emph{[[fixed point spectrum]]} $F^G(E)$ and its \emph{geometric fixed point spectrum} $\Phi^G(E)$. Here the terminology ``geometric'' is in the sense of \emph{[[point-set topology]]}, as opposed to [[homotopy theory]]: If $X$ is a ([[pointed topological space|pointed]]) [[topological space]] equipped with a [[continuous function]] $G$-[[action]] (a [[topological G-space]]), so that one may consider its $G$-[[equivariant suspension spectrum]] $\Sigma^\infty_G X \in G Spectra$, then the \emph{geometric fixed point spectrum} $\Phi^G(\Sigma^\infty_G X)$ of the latter is equivalent to the plain [[suspension spectrum]] of the plain [[fixed point]]-space $X^G$: \begin{displaymath} \Phi^G\big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty\big( X^G \big) \,, \end{displaymath} see the characterization in Prop. \ref{AbstractCharacterizationOfGeometricFixedPointSpectra}, below. In general this is different from (not [[equivalence in an (infinity,1)-category|equivalent]] to) both of the following other notions of fixed point spectra: \begin{enumerate}% \item the plain (really: [[homotopy theory|homotopy theoretic]]) [[fixed point spectrum]] $F^G(\Sigma^\infty_G X)$, which is instead the [[derived functor]] of forming topological fixed points $X \mapsto X^G$, hence which applies this construction only after [[fibrant resolution]]; the difference between the two is described by the \emph{[[tom Dieck splitting theorem]]}, see Prop. \ref{AsWedgeSummandInTomDieckSplitting} below. \item the [[categorical fixed point spectrum]]\ldots{} \end{enumerate} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} A concrete definition of geometric fixed points of an equivariant spectrum is in (\hyperlink{Schwede15}{Schwede 15, 7.3}). A conceptual characterization in terms of [[localization of spectra]] is in (\hyperlink{MathewNaumannNoel15}{Mathew-Naumann-Noel 15, def. 6.12}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{ForEquivariantSuspensionSpectra}{}\subsubsection*{{For equivariant suspension spectra}}\label{ForEquivariantSuspensionSpectra} \begin{prop} \label{AsWedgeSummandInTomDieckSplitting}\hypertarget{AsWedgeSummandInTomDieckSplitting}{} \textbf{(as a [[wedge sum|wedge summand]] in the [[tom Dieck splitting]])} For $X$ a [[topological G-space]] and $\Sigma^\infty_G X$ its [[equivariant suspension spectrum]], there is a canonical comparison morphism (\ldots{}) \begin{displaymath} \Phi^G(\Sigma^\infty_G X) \;\simeq\; \Sigma^\infty( X^G ) \hookrightarrow F^G(\Sigma^\infty_G X) \end{displaymath} which exhibits its geometric fixed point spectrum as precisely the first summand in the [[tom Dieck splitting]] of the plain [[fixed point spectrum]] \begin{displaymath} F^G(\Sigma^\infty_G X) \simeq \Sigma^\infty( X^G ) \vee \left( \underset{{[H\subset G]} \atop {1 \neq H \neq G}}{\vee} \Sigma^\infty( E (W_G H)_+ \wedge_{W_G H} X^H ) \right) \vee \Sigma^\infty( E G_+ \wedge_{G} X ) \,. \end{displaymath} \end{prop} (\hyperlink{Schwede15}{Schwede 15, Example 7.7}) In fact: \begin{prop} \label{AbstractCharacterizationOfGeometricFixedPointSpectra}\hypertarget{AbstractCharacterizationOfGeometricFixedPointSpectra}{} The construction of geometric fixed point spectra is essentially uniquely characterized by the property \begin{displaymath} \Phi^G\big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty\big( X^G \big) \end{displaymath} and the property of being [[left derived functor|left derived]] [[strong monoidal functor|strong monoidal]] and preserving [[homotopy colimits]]. \end{prop} (\hyperlink{Schwede15}{Schwede 15, remark 7.15}, \hyperlink{Blumberg17}{Blumberg 17, around Def. 2.5.16}) More generally: \begin{prop} \label{PartialGeometricFixedPoint}\hypertarget{PartialGeometricFixedPoint}{} \textbf{(partial geometric fixed point spectra)} There is a ``partial'' geometric fixed point functor, which for a given [[subgroup]] $H \subset G$ sends \begin{displaymath} \Phi^H \;\colon\; G Spectra \longrightarrow W_G H Spectra \end{displaymath} (for $W_G/H$ the [[Weyl group]], which is the [[quotient group]] $G/H$ in the case that $H$ is a [[normal subgroup]]) and satisfies for a $G$-[[equivariant suspension spectrum]] $\Sigma^\infty_G X$ the relation \begin{equation} \Phi^N \big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty_{W_G H} X^H \,, \label{PartialGeometricFixedPointsOfEqui}\end{equation} hence, if $H = N \subset G$ already happens to be a [[normal subgroup]]: \begin{displaymath} \Phi^N \big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty_{G/N} X^H \,. \end{displaymath} \end{prop} (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, II.9, Def. 9.7, Cor. 9.9}, \hyperlink{Lewis00}{Lewis 00, Scholium 10.2}) $\,$ \hypertarget{InTermsOfSmashingLocalization}{}\subsubsection*{{In terms of smashing localization}}\label{InTermsOfSmashingLocalization} We collect some facts from \hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, section II.9}. Throughout, consider a [[finite group]] $G$ and a [[normal subgroup]] $N \subset G$. \begin{defn} \label{}\hypertarget{}{} We write \begin{displaymath} \mathcal{F}[N] \;\coloneqq\; \big\{ N ⊄ H\; \subset G \big\} \end{displaymath} for the set of [[subgroups]] of $G$ that do not contain $N$, and \begin{displaymath} \mathcal{F}[N]^' \;\coloneqq\; \big\{ N \subset H \subset G \big\} \end{displaymath} for the subset of [[subgroups]] of $G$ that do contain $N$. \end{defn} (\hyperlink{LewisMaySteinberger86}{LMS86, p. 107 \& bottom of p. 109}) \begin{defn} \label{tildeE}\hypertarget{tildeE}{} There is a [[pointed topological space|pointed]] [[G-space]] \begin{displaymath} \widetilde E \mathcal{F}[N] \;\in\; G Spaces \end{displaymath} whose [[fixed point spaces]] for [[subgroups]] $H \subset G$ are \begin{displaymath} \big( \widetilde E \mathcal{F}[N] \big)^H \;\simeq\; \left\{ \itexarray{ \ast &\vert& H \in \mathcal{F}[N] \;\Leftrightarrow\; N ⊄ H \\ S^0 &\vert& H \in \mathcal{F}[N]^' \;\Leftrightarrow\; N \subset H } \right. \end{displaymath} \end{defn} (\hyperlink{LewisMaySteinberger86}{LMS86, beginning of II.9}) \begin{defn} \label{FPrimeEquivalences}\hypertarget{FPrimeEquivalences}{} We say that a morphism $f \colon X \to Y$ of [[G-spectra]] is an \emph{$\mathcal{F}[N]^'$-equivalence} if its [[smash product]] with $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE}) \begin{displaymath} f \wedge id_{\tilde E \mathcal{F}[N]} \;\colon\; X \wedge \tilde E \mathcal{F}[N] \longrightarrow Y \wedge \tilde E \mathcal{F}[N] \end{displaymath} is an equivalence of [[G-spectra]]. \end{defn} \hyperlink{LewisMaySteinberger86}{LMS 86, bottom of p. 107} \begin{prop} \label{FPrimeLocalizationIsSmashingLocalization}\hypertarget{FPrimeLocalizationIsSmashingLocalization}{} The [[localization]] of $G Spectra$ at the $\mathcal{F}[N]'$-equivalences (Def. \ref{FPrimeEquivalences}) is a [[smashing localization]], given by smashing with the [[equivariant suspension spectrum]] of $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE}) \begin{displaymath} \Sigma^\infty_G \tilde E \mathcal{F}[N] \;\in\; G Spectra \end{displaymath} In particular, we have \begin{displaymath} Ho_{G Spectra}\left( \tilde E \mathcal{F}[N] \wedge X, \tilde E \mathcal{F}[N] \wedge Y \right) \;\simeq\; Ho_{G Spectra}\left( X, \tilde E \mathcal{F}[N] \wedge Y \right) \end{displaymath} \end{prop} (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, Prop. II 9.1, 9.2 \& top of p. 109}) \begin{remark} \label{LocalizationBySmashingWithtildeEF}\hypertarget{LocalizationBySmashingWithtildeEF}{} Hence \begin{displaymath} (L_{\mathcal{F}[N]'})_{X,Y} \coloneqq Ho_{G Spectra}(X, (S^0 \to \tilde E \mathcal{F}) \wedge Y) \;\colon\; Ho_{G Spectra}(X,Y) \longrightarrow Ho_{L_{\mathcal{F}[N]'} G Spectra}(X,Y) \end{displaymath} is $\mathcal{F}'[N]$-localization on [[hom-objects]]. \end{remark} \begin{lemma} \label{SmashingWithTildeOnSpaces}\hypertarget{SmashingWithTildeOnSpaces}{} For $X$ and $Y$ [[G-CW-complexes]], the following are [[bijections]] of [[hom-sets]]: \begin{displaymath} \itexarray{ Ho_{G Spaces)} \big( X, \widetilde E \mathcal{F}[N] \wedge Y \big) & \underoverset{\simeq}{ Ho_{G Spaces} \big( X^N \hookrightarrow X, \widetilde E \mathcal{F}[N] \big) }{\longrightarrow} & \mathrm{Hom}_{Ho(G Spaces)} \big( X^N, \widetilde E \mathcal{F}[N] \wedge Y \big) \\ && = \\ Ho_{G Spaces} \big( X^N, Y \big) & \underoverset{\simeq}{ Ho_{G Spaces} \big( X^N, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y \big) }{\longrightarrow} & Ho_{G Spaces} \big( X^N, \widetilde E \mathcal{F}[N] \wedge Y \big) \\ = \\ Ho_{G Spaces} \big( X^N, Y^N \big) } \end{displaymath} \end{lemma} (\hyperlink{LewisMaySteinberger86}{LMS 86, prop. II 9.3 with remark below the proof}) \begin{cor} \label{SmashingWithTildeEOnspacesEquivalentToRestriction}\hypertarget{SmashingWithTildeEOnspacesEquivalentToRestriction}{} On [[hom-sets]] of [[G-spaces]] $Ho_{G Spaces}(X,Y)$, [[postcomposition|postcomposing]] with the [[smash product|smashing]] $(S^0 \to \tilde E \mathcal{F}[N]) \wedge Y$ is isomorphic to restricting along $X^N \hookrightarrow X$: The following is a [[commuting square]] (by nature of the [[hom-functor]]) and the right and bottom morphisms are bijections by Lemma \ref{SmashingWithTildeOnSpaces}: \begin{displaymath} \itexarray{ Ho_{G Spaces} \big( X, Y \big) & \overset{ Ho_{G Spaces} \big( X^N \hookrightarrow N, Y \big) }{ \longrightarrow } & Ho_{G Spaces}\big( X^N, Y \big) &\simeq& Ho_{G Spaces}\big( X^N, Y^N \big) \\ {}^{ \mathllap{ Ho_{G Spaces}( X, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y ) } } \big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow {}^{ \mathrlap{ Ho_{G Spaces}( X^N, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y ) } } \\ Ho_{G Spaces}\big( X, \tilde E \mathcal{F}[N] \wedge Y \big) & \underoverset {\simeq} { Ho_{G Spaces} \big( X^N \hookrightarrow X, \tilde E \mathcal{F}[N] \wedge Y \big) } {\longrightarrow} & Ho_{G Spaces}\big( X^N, \tilde E \mathcal{F}[N] \wedge Y \big) } \end{displaymath} \end{cor} \begin{lemma} \label{GeometricFixedPointsInTermsOfPlainFixedPoints}\hypertarget{GeometricFixedPointsInTermsOfPlainFixedPoints}{} \textbf{(geometric fixed point spectra in terms of homotopy fix point spectra)} The partial geometric fixed point functor (Prop. \ref{PartialGeometricFixedPoint}) \begin{displaymath} \Phi^N \;\colon\; G Spectra \longrightarrow G/N Spectra \end{displaymath} is given on [[equivariant suspension spectra]] $\Sigma^\infty_G X$ equivalently by first smashing with $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE}) followed by forming the partial plain [[fixed point spectrum]]: \begin{displaymath} \Phi^N \Sigma^\infty_G X \;\simeq\; \big( \tilde E \mathcal{F}[N] \;\wedge\; \Sigma^\infty_G X \big)^N \,. \end{displaymath} \end{lemma} (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, Cor. 9.9}) We will also need this here: \begin{lemma} \label{FixedPointSpectrumOfSmashProduct}\hypertarget{FixedPointSpectrumOfSmashProduct}{} For $X$ a [[G-CW-complex]] $E$ a [[G-spectrum|G-]] [[CW-spectrum]] and $N \subset G$ a [[normal subgroup]], the partial $N$-[[fixed point spectrum]] functor on spectra and the plain fixed point functor on spaces are compatible with smash product: \begin{displaymath} \left( \tilde E \mathcal{F}[N] \wedge E \wedge X \right)^N \;\simeq\; \left( \tilde E \mathcal{F}[N] \wedge E \right) \wedge X^N \end{displaymath} \end{lemma} (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, prop. II 9.12}) \hypertarget{AsInversionOfEulerClasses}{}\subsubsection*{{Via inversion of Euler classes}}\label{AsInversionOfEulerClasses} We discuss an explicit formula (Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} below, due to \hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86}) that expresses [[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in partial geometric fixed point spectra (Prop. \ref{PartialGeometricFixedPoint}) as the [[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in the original spectrum, but with certain ``[[Euler classes]] [[localization|inverted]]''. As an application, we show (Example \ref{EquivariantStableCohomotopyOfPointsInNontrivalROGDegree} below) that the [[equivariant stable cohomotopy]] of the point in certain non-trivial [[RO(G)-degrees]] $V$ surjects onto the corresponding partially equivariant stable cohomotopy in degree 0 (the latter being well-understood: given by the [[Burnside ring]], by [[Burnside ring is equivariant stable cohomotopy of the point|this Prop]]). $\,$ A key role in this discussion is played by those [[RO(G)-degrees]] which trivialize when passing to partial fixed points: \begin{defn} \label{ROGDegreeWithoutNonTrivialHFixedPoints}\hypertarget{ROGDegreeWithoutNonTrivialHFixedPoints}{} \textbf{([[RO(G)-degrees]] without non-trivial $H$-fixed points)} For $H \subset G$ a [[subgroup]], say that a $G$-[[representation]] $V$ has \emph{no non-trivial $H$-fixed points} if the [[fixed point space]] of $V$ with respect to the $H$-[[action]] is the origin (which is necessarily fixed), hence is the [[zero object|zero]]-[[representation]]: \begin{displaymath} V^H \;=\; 0 \end{displaymath} \end{defn} We also use the following notation, following \hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86}: \begin{defn} \label{}\hypertarget{}{} \textbf{([[base change]] along [[normal subgroup]]-inclusions of [[equivariant homotopy theory|equivariance]]-groups)} Given a [[normal subgroup]]-inclusion \begin{displaymath} N \overset{\phantom{AA}}{\hookrightarrow} G \overset{ \phantom{A} \epsilon \phantom{A} }{\longrightarrow} G/N \end{displaymath} with induced [[projection]] $\epsilon$ to the [[quotient group]] $G/N$ this induces various [[base change]] [[adjunctions]] (on [[homotopy categories]], say), such as on [[topological G-spaces]], to be denoted \begin{equation} G/N Spaces \overset{ \epsilon^\sharp }{\longrightarrow} G Spaces \label{PullbackGSpace}\end{equation} and on $G$-[[representations]], to be denoted \begin{equation} G/N Rep { \overset{ \epsilon^\ast }{\longrightarrow} } G Rep \label{PullbackRepresentation}\end{equation} and on [[G-spectra]], to be denoted \begin{equation} G/N Spectra \underoverset { \underset{(-)^N}{\longleftarrow} } { \overset{ \epsilon^\sharp }{\longrightarrow} } {\phantom{AA}\bot\phantom{AA}} G Spectra \label{PullbackSpectrum}\end{equation} where the [[right adjoint]] $(-)^N$ is the partial [[fixed point spectrum]]-functor (in contrast to the \emph{geometric} fixed point functor). \end{defn} (e.g. \hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, above theorem 9.5}) \begin{prop} \label{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses}\hypertarget{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses}{} \textbf{(partial geometric fixed point cohomology via inversion of Euler classes)} Let $E \;\in\; G Spectra$ be a [[G-spectrum]] with partial geometric fixed point spectrum $\Phi^N E \;\in\; G/N Spectra$ (Prop. \ref{PartialGeometricFixedPoint}) and let $X \;\in\; G/N Spectra^{fin} \overset{\epsilon^\sharp}{\longrightarrow} G Spectra$ be [[finite spectrum|finite]] $G$-[[CW-spectrum]]. Then the $G/N$-[[equivariant cohomology|equivariant]] [[cohomology groups]] in [[RO(G)-degree|RO(G/N)-degree]] $\alpha$ of $X$ with [[coefficients]] in the partial geometric fixed point spectrum $\Phi^N E$ are equivalently the [[colimit]] over the $G$-[[equivariant cohomology|equivariant]] [[cohomology groups]] of $\epsilon^\sharp X$ \eqref{PullbackGSpace} with [[coefficients]] in $E$, but in [[RO(G)-degree]] $\epsilon^\ast \alpha$ \eqref{PullbackRepresentation} plus a shift by all those [[representations]] $V$ that have no nontrivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}): \begin{equation} (\Phi^N E)^\bullet_{G/N}(X) \;\simeq\; \underset{\underset{\{V \vert V^N = 0\}}{\longrightarrow}}{\lim} E_G^{\epsilon^\ast \bullet + V}(\epsilon^\sharp X) \,, \label{ColimitConstructionForCohomologyWithCoeffsInPartialGeometricFixedPointSpectra}\end{equation} where the [[colimit]] is over the [[diagram]] that has precisely one morphism for every inclusion $V_1 \subset V_2$ of $G$-representations without non-trivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}) \begin{displaymath} E_G^{\epsilon^\ast \bullet + V_1}(\epsilon^\sharp X) \overset{ (-) \wedge \chi_{V_2 - V_1} }{\longrightarrow} E_G^{\epsilon^\ast \bullet + V_2}(\epsilon^\sharp X) \end{displaymath} given by [[smash product]] with the [[Euler class]] \begin{displaymath} \chi_{V} \;\coloneqq\; 1 \in E_G^V(S^V) \overset{ (S^0 \to S^V)^\ast }{\longrightarrow} E_G^V(S^0) \end{displaymath} of $V \coloneqq V_2 - V_1$. \end{prop} (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, chapter II, prop. 9.13}) \begin{prop} \label{CanonicalComparisonMap}\hypertarget{CanonicalComparisonMap}{} \textbf{(comparison map to partial geometric fixed point cohomology)} Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} provides a canonical comparison morphism, to be denoted \begin{equation} E_G^{\epsilon^\ast \bullet}(\epsilon^\sharp X) \overset{ \phantom{AA} p_E^N(X) \phantom{AA} }{\longrightarrow} (\Phi^N E)^\bullet_{G/N}(X) \label{ComparisonMap}\end{equation} from the $G$-[[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in $E$ to those with [[coefficients]] in the partial geometric fixed point spectrum: Namely the component of the [[colimit|colimiting]] [[cocone]]\eqref{ColimitConstructionForCohomologyWithCoeffsInPartialGeometricFixedPointSpectra} at stage $V = 0$: \begin{displaymath} \itexarray{ E^{\epsilon^\ast \bullet }(\epsilon^\sharp X) && \overset{ (-) \wedge \chi_{V} }{\longrightarrow} && E^{\epsilon^\ast \bullet + V}(\epsilon^\sharp X) &\to& \cdots \\ & {}_{ \mathllap{ p_E^N(X) } }\searrow && \swarrow & \cdots \\ && (\Phi^N E)^\bullet_{G/N}(X) } \end{displaymath} This component is equal to the following composite of isomorphisms with $\mathcal{F}[N]'$-localization $L_{\mathcal{F}[N]'}$ (Def. \ref{FPrimeEquivalences}): \begin{equation} \begin{aligned} p_E^N(X) \;\colon\; E^{\epsilon^\ast \alpha}(\epsilon^\sharp X) & = Ho_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge E \right) \\ & \simeq Ho_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge S^0 \wedge E \right) \\ & \overset{ L_{\mathcal{F}[N]'} }{\longrightarrow} Ho_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge \tilde E \mathcal{F}[N]\wedge E \right) \\ & \simeq Ho_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; \left( \Sigma^\infty_G S^{\epsilon^\ast \alpha} \wedge \tilde E \mathcal{F}[N] \wedge E \right)^N \right) \\ & \simeq Ho_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; \left( S^{\epsilon^\ast \alpha} \right)^N \wedge \left( \tilde E \mathcal{F}[N] \wedge E \right)^N \right) \\ & \simeq Ho_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; S^{\alpha} \wedge \Phi^N E \right) \\ & = (\Phi^N E)^{\alpha}(X) \end{aligned} \label{LocalizationIsComparisonMorphism}\end{equation} \end{prop} \begin{proof} This follows from the proof of (\hyperlink{LewisMaySteinberger86}{Lewis-May-Steinberger 86, chapter II, prop. 9.13}). We make this explicit: The proof there says that the comparison map is given by the [[smashing localization|smashing]] with $S^0 \to \tilde E \mathcal{F}$, up to re-identifications: \begin{enumerate}% \item The first equality in \eqref{LocalizationIsComparisonMorphism} is the definition of cohomology classes; \item the second step is the [[unitor]] isomorphism for the [[tensor unit]] being the [[sphere spectrum]]; \item the third step is smashing with $S^0 \to \tilde E \mathcal{F}[N]$, which is $\mathcal{F}'$-localization by Prop. \ref{FPrimeLocalizationIsSmashingLocalization}; \item the fourth step is the hom-isomorphism for the [[adjoint functor|adjunction]] $( \epsilon^\sharp \dashv (-)^N )$ from \eqref{PullbackSpectrum}; \item the fifth step is application of Lemma \ref{FixedPointSpectrumOfSmashProduct}; \item the sixth step is the evident identification $(S^{\epsilon^\ast \alpha})^N = S^\alpha$ in the first smash factor, and is Lemma \ref{GeometricFixedPointsInTermsOfPlainFixedPoints} in the second factor. \item the seventh step is again the definition of cohomology. \end{enumerate} \end{proof} $\,$ \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{EquivariantStableCohomotopyOfPointsInNontrivalROGDegree}\hypertarget{EquivariantStableCohomotopyOfPointsInNontrivalROGDegree}{} \textbf{([[equivariant stable cohomotopy]] of the point in non-trivial [[RO(G)-degree]])} Let $G$ be a [[finite group]]. Then the canonical comparison morphism \eqref{ComparisonMap} from Def. \ref{CanonicalComparisonMap} exhibits the $G$-[[equivariant stable cohomotopy]] [[cohomology groups|group]] of the point in any [[RO(G)-degree]] $V$ that has trivial $N$-fixed points ($V^N = 0$, Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}) as a [[group extension]] of the $G/N$-[[equivariant stable cohomotopy]] of the point in [[RO(G)-degree|RO(G/N)-degree]] zero, hence of the group underlying the [[Burnside ring]] $A(G/N)$ ([[Burnside ring is equivariant stable cohomotopy of the point|this Prop.]]): \begin{equation} \mathbb{S}_G^{V}(\ast) \overset { \phantom{A} \text{epi} \phantom{A} } {\longrightarrow} \mathbb{S}_{G/N}^0(\ast) \simeq A(G/N) \,. \label{SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero}\end{equation} \end{example} \begin{proof} First observe that, in the given situation, the comparison morphism $p_{\mathbb{S}}^N(\ast)$ \eqref{ComparisonMap} is indeed of the form shown, up to isomorphism: We are in the situation of Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} for \begin{enumerate}% \item $X \coloneqq \Sigma^\infty_{G/N}(\ast_+) = \Sigma^\infty_{G/N} S^0$, which is clearly a [[finite spectrum|finite]] $G/N$-[[CW-spectrum]]; \item $E \coloneqq \Sigma^V_G\mathbb{S}_G \coloneqq \Sigma^\infty_G S^V$ the $V$-shifted $G$-[[equivariant sphere spectrum]], being the [[G-spectrum]] [[Brown representability theorem|representing]] $G$-[[equivariant stable cohomotopy]], by definition; \item $\Phi^N E \simeq \Sigma^\infty_{G/N} (S^V)^N \simeq \Sigma^\infty_{G/N} S^0 \simeq \mathbb{S}_{G/N}$ the unshifted $G/N$-[[equivariant sphere spectrum]], by \eqref{PartialGeometricFixedPointsOfEqui} and by assumption on $V$. \end{enumerate} Hence with all identifications made explicit, the morphism \eqref{SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero} in question is the composite \begin{equation} \mathbb{S}_G^V(\ast) \simeq (\Sigma^\infty_G S^V)^0_{G}(\ast) \overset{ \phantom{AA} p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) \phantom{AA} }{\longrightarrow} (\Phi^N \Sigma^\infty_G S^V)^0_{G/N}(\ast) \simeq (\Sigma^\infty_{G/N} S^0)^0_{G/N}(\ast) \simeq \mathbb{S}^0_{G/N}(\ast) \simeq A(G/N) \label{ProjectionFromEquivariantCohomotopyOfPpintInRODegreeToBurnside}\end{equation} of $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ with a sequence of [[isomorphisms]], and hence our task is to prove that $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is a surjection. We first prove this for the case that $V = 0$. In this case the identification with the [[Burnside ring]] (via [[Burnside ring is equivariant stable cohomotopy of the point|this Prop.]]) applies also to the [[domain]] cohomology group: \begin{displaymath} \mathbb{S}_G^V(\ast) \;\simeq\; \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big(S^V, S^V \big) \simeq A(G) \,, \end{displaymath} By Prop. \ref{CanonicalComparisonMap} the comparison morphism acts on this by smashing the [[codomain]] of the [[hom-sets]] with $(S^0 \to \tilde E \mathcal{F}[N])$. But by Corollary \ref{SmashingWithTildeEOnspacesEquivalentToRestriction} this is equivalent to restricting to $N$-[[fixed point spaces]] so that \eqref{SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero} becomes simply the projection of [[Burnside rings]] \begin{displaymath} \itexarray{ A(G) &\simeq& \underset{\underset{ V \in G Rep }{\longrightarrow}}{\lim} & Ho_{G Spaces}\big( S^V, S^V \big) &\simeq& \mathbb{S}^0_G(\ast) \\ {}^{\mathllap{ (-)^N }}\big\downarrow && \big\downarrow {}^{{}_{ \mathrlap{ \itexarray{ \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big( S^V, S^V \wedge (S^0 \to \tilde E\mathcal{F}[E]) \big) \\ = \\ \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big( (S^V)^N \hookrightarrow S^V, S^V \big) } } } } & && \big\downarrow{}^{ \mathrlap{ p_\mathbb{S}^N(\ast) } } \\ A(G/N) &\simeq& \underset{\underset{ W \in G/N Rep }{\longrightarrow}}{\lim} & Ho_{G/N Spaces}\big( S^W, S^W \big) &\simeq& \mathbb{S}^0_{G/N}(\ast) } \end{displaymath} sending any [[G-set]] $K$ to its [[subset]] $K^N$ of $N$-[[fixed points]] regarded with its residual $G/N$-[[action]]. This is clearly surjective. (The irreducible elements on the right are the isomorphism classes of the [[transitive action|transitive]] $G/N$-[[actions]] $(G/N)/H$ for $H \subset G/H$, which are canonically also [[G-sets]], hence have a pre-image on the left.) In order to deduce the general statement from this special case, we now make use of the fact that Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} says that the comparison map for $V = 0$ is one coprojection map of a [[colimit|colimiting]] [[cocone]]-[[diagram]], which for each $G$-[[representation]] $V$ without non-trivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}) contains a [[cocone]] component of the following form: \begin{equation} \itexarray{ A(G) &=& (\Sigma^\infty_G S^0)^0_{G}(\ast) &\overset{ (-) \wedge \chi_V }{\longrightarrow}& (\Sigma^\infty_G S^V)^0_{G}(\ast) &\to& \cdots \\ {}^{(-)^N}\big\downarrow && {}^{\mathllap{epi}}\big\downarrow{}^{p_0} & \swarrow_{\mathrlap{p_V}} \\ A(G/N) &=& (\Sigma^\infty_{G/N} S^0)^0_{G/N}(\ast) } \label{ColimitingCoconeForComparisonAtTrivialROG}\end{equation} Since we know, as just argued, that the map $(-)^N$ on the left is a surjection, the [[commutative diagram|commutativity]] of this diagram implies that also the component projection $p_V$ is surjective. (Every element $c \in A(G/N)$ has a lift to $\widehat c \in A(G)$, but then the commutativity of the triangle means that $\widehat c \wedge \chi_V$ is a pre-image of $c$ under $p_V$.) This we may use to deduce the statement for the general case, where the codomain of \eqref{ProjectionFromEquivariantCohomotopyOfPpintInRODegreeToBurnside} is in degree $V$: By the assumption that the [[RO(G)-degree]] $V$ has no non-trivial $N$-fixed points, Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} says that the [[colimit|colimiting]] [[cocone]] in which the map $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ appears, by Def. \ref{CanonicalComparisonMap}, looks just like the one above, except that it ``starts'' not in degree 0, but in degree $V$: \begin{equation} \itexarray{ && (\Sigma^\infty_G S^V)^0_{G}(\ast) &\to& \cdots \\ & \swarrow_{\mathrlap{p_V}} \\ (\Sigma^\infty_G S^0)^0_{G/N}(\ast) } \label{ColimitingCoconeForComparisonAtNonTrivialROG}\end{equation} In particular the [[cocone]] in \eqref{ColimitingCoconeForComparisonAtTrivialROG} restricts to a [[cocone]] over this sub-diagram in \eqref{ColimitingCoconeForComparisonAtNonTrivialROG}, so that the [[universal property]] of the cocone in \eqref{ColimitingCoconeForComparisonAtNonTrivialROG} implies an [[endomorphism]] $\phi$ of the [[abelian group]] underlying the [[Burnside ring]] $(\Sigma^\infty_G S^0)^0_{G/N}(\ast) = A(G/n)$ such that \begin{equation} p_V \;=\; \phi \;\circ\; p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) \,. \label{Factorization}\end{equation} Since $p_V$ is surjective, it is now sufficient to prove that this $\phi$ is in fact an [[isomorphism]]. To see this, observe that, since $G$ is a [[finite group]] by assumption, the [[abelian group]] underlying the [[Burnside ring]] $A(G/N)$ is a [[finitely generated module|finitely generated]] [[free abelian group]] (spanned by the cosets $(G/N)/H$ as $H$ ranges over the [[finite set]] of [[conjugacy classes]] of [[subgroups]] of $G/N$ ). By the \href{principal+ideal+domain#StructureTheoryOfModules}{structure theory of free abelian groups}, this means that $\phi$ may be represented by a [[matrix]] in [[Smith normal form]]. Specifically, since $\phi$ is an [[endomorphism]], it is represented by a square matrix in [[Smith normal form]]. Since $\phi \circ p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective, by \eqref{Factorization} and the surjectivity of $p_V$ established before, this implies that $\phi$ is represented by a [[diagonal matrix]] all whose diagonal entries are non-vanishing and invertible, hence that $\phi$ is in fact an [[isomorphism]]. With this, \eqref{Factorization} says that with $p_V$ also $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective. \end{proof} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[fixed point space]] \item [[fixed point spectrum]] \item [[categorical fixed point spectrum]] \item [[homotopy fixed points]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics Vol.1213. 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf}) \item [[L. Gaunce Lewis, Jr.]], section 10 of \emph{Splitting theorems for certain equivariant spectra}, Memoirs of the AMS, number 686, March 2000, Volume 144 (\href{http://hopf.math.purdue.edu/LewisG/spltspec.pdf}{pdf}) \item [[Stefan Schwede]], section 7.3 of \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf}) \item [[Stefan Schwede]], section 3.3. of \emph{[[Global homotopy theory]]} (\href{https://arxiv.org/abs/1802.09382}{arXiv:1802.09382}) \item [[Andrew Blumberg]], Def. 2.5.16 in \emph{Equivariant homotopy theory}, 2017 (\href{https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf}{pdf}, \href{https://github.com/adebray/equivariant_homotopy_theory}{GitHub}) \item [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], \emph{Nilpotence and descent in equivariant stable homotopy theory} (\href{http://arxiv.org/abs/1507.06869}{arXiv:1507.06869}) \item [[Tom Bachmann]], [[Marc Hoyois]], remark 9.9 in \emph{Norms in motivic homotopy theory} (\href{https://arxiv.org/abs/1711.03061}{arxiv:1711.03061}) \end{itemize} Relation to [[spectral Mackey functors]]: \begin{itemize}% \item [[Saul Glasman]], \emph{Stratified categories, geometric fixed points and a generalized Arone-Ching theorem} (\href{https://arxiv.org/abs/1507.01976}{arXiv:1507.01976}, \href{http://www-users.math.umn.edu/~sglasman/strattalk.pdf}{talk notes pdf}) \end{itemize} [[!redirects geometric fixed point spectra]] [[!redirects geometric fixed point]] [[!redirects geometric fixed points]] [[!redirects geometric fixed point functor]] [[!redirects geometric fixed point functors]] \end{document}