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geometric fixed point spectrum

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

In the context of equivariant stable homotopy theory and GG one distinguishes, for a G-spectrum EE, between the plain fixed point spectrum F G(E)F^G(E) and its geometric fixed point spectrum Φ G(E)\Phi^G(E).

Here the terminology “geometric” is in the sense of point-set topology, as opposed to homotopy theory: If XX is a (pointed) topological space equipped with a continuous function GG-action (a topological G-space), so that one may consider its GG-equivariant suspension spectrum Σ G XGSpectra\Sigma^\infty_G X \in G Spectra, then the geometric fixed point spectrum Φ G(Σ G X)\Phi^G(\Sigma^\infty_G X) of the latter is equivalent to the plain suspension spectrum of the plain fixed point-space X GX^G:

Φ G(Σ G X)Σ (X G), \Phi^G\big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty\big( X^G \big) \,,

see the characterization in Prop. , below.

In general this is different from (not equivalent to) both of the following other notions of fixed point spectra:

  1. the plain (really: homotopy theoretic) fixed point spectrum F G(Σ G X)F^G(\Sigma^\infty_G X), which is instead the derived functor of forming topological fixed points XX GX \mapsto X^G, hence which applies this construction only after fibrant resolution; the difference between the two is described by the tom Dieck splitting theorem, see Prop. below.

  2. the categorical fixed point spectrum?

Definition

A concrete definition of geometric fixed points of an equivariant spectrum is in (Schwede 15, 7.3). A conceptual characterization in terms of localization of spectra is in (Mathew-Naumann-Noel 15, def. 6.12).

Properties

For equivariant suspension spectra

Proposition

(as a wedge summand in the tom Dieck splitting)

For XX a topological G-space and Σ G X\Sigma^\infty_G X its equivariant suspension spectrum, there is a canonical comparison morphism (…)

Φ G(Σ G X)Σ (X G)F G(Σ G X) \Phi^G(\Sigma^\infty_G X) \;\simeq\; \Sigma^\infty( X^G ) \hookrightarrow F^G(\Sigma^\infty_G X)

which exhibits its geometric fixed point spectrum as precisely the first summand in the tom Dieck splitting of the plain fixed point spectrum

F G(Σ G X)Σ (X G)([HG]1HGΣ (E(W GH) + W GHX H))Σ (EG + GX). 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 ) \,.

(Schwede 15, Example 7.7)

In fact:

Proposition

The construction of geometric fixed point spectra is essentially uniquely characterized by the property

Φ G(Σ G X)Σ (X G) \Phi^G\big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty\big( X^G \big)

and the property of being left derived strong monoidal and preserving homotopy colimits.

(Schwede 15, remark 7.15, Blumberg 17, around Def. 2.5.16)

More generally:

Proposition/Remark

(partial geometric fixed point spectra)

There is a “partial” geometric fixed point functor, which for a given subgroup HGH \subset G sends

Φ H:GSpectraW GHSpectra \Phi^H \;\colon\; G Spectra \longrightarrow W_G H Spectra

(for W G/HW_G/H the Weyl group, which is the quotient group G/HG/H in the case that HH is a normal subgroup) and satisfies for a GG-equivariant suspension spectrum Σ G X\Sigma^\infty_G X the relation

(1)Φ N(Σ G X)Σ W GH X H, \Phi^N \big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty_{W_G H} X^H \,,

hence, if H=NGH = N \subset G already happens to be a normal subgroup:

Φ N(Σ G X)Σ G/N X H. \Phi^N \big( \Sigma^\infty_G X \big) \;\simeq\; \Sigma^\infty_{G/N} X^H \,.

(Lewis-May-Steinberger 86, II.9, Def. 9.7, Cor. 9.9, Lewis 00, Scholium 10.2)

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In terms of smashing localization

We collect some facts from Lewis-May-Steinberger 86, section II.9.

Throughout, consider a finite group GG and a normal subgroup NGN \subset G.

Definition

We write

[N]{NHG} \mathcal{F}[N] \;\coloneqq\; \big\{ N ⊄ H\; \subset G \big\}

for the set of subgroups of GG that do not contain NN, and

[N] {NHG} \mathcal{F}[N]^' \;\coloneqq\; \big\{ N \subset H \subset G \big\}

for the subset of subgroups of GG that do contain NN.

(LMS86, p. 107 & bottom of p. 109)

Definition

There is a pointed G-space

E˜[N]GSpaces \widetilde E \mathcal{F}[N] \;\in\; G Spaces

whose fixed point spaces for subgroups HGH \subset G are

(E˜[N]) H{* | H[N]NH S 0 | H[N] NH \big( \widetilde E \mathcal{F}[N] \big)^H \;\simeq\; \left\{ \array{ \ast &\vert& H \in \mathcal{F}[N] \;\Leftrightarrow\; N ⊄ H \\ S^0 &\vert& H \in \mathcal{F}[N]^' \;\Leftrightarrow\; N \subset H } \right.

(LMS86, beginning of II.9)

Definition

We say that a morphism f:XYf \colon X \to Y of G-spectra is an [N] \mathcal{F}[N]^'-equivalence if its smash product with E˜[N]\tilde E \mathcal{F}[N] (Def. )

fid E˜[N]:XE˜[N]YE˜[N] f \wedge id_{\tilde E \mathcal{F}[N]} \;\colon\; X \wedge \tilde E \mathcal{F}[N] \longrightarrow Y \wedge \tilde E \mathcal{F}[N]

is an equivalence of G-spectra.

LMS 86, bottom of p. 107

Proposition

The localization of GSpectraG Spectra at the [N]\mathcal{F}[N]'-equivalences (Def. ) is a smashing localization, given by smashing with the equivariant suspension spectrum of E˜[N]\tilde E \mathcal{F}[N] (Def. )

Σ G E˜[N]GSpectra \Sigma^\infty_G \tilde E \mathcal{F}[N] \;\in\; G Spectra

In particular, we have

Ho GSpectra(E˜[N]X,E˜[N]Y)Ho GSpectra(X,E˜[N]Y) 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)

(Lewis-May-Steinberger 86, Prop. II 9.1, 9.2 & top of p. 109)

Remark

Hence

(L [N]) X,YHo GSpectra(X,(S 0E˜)Y):Ho GSpectra(X,Y)Ho L [N]GSpectra(X,Y) (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)

is [N]\mathcal{F}'[N]-localization on hom-objects.

Lemma

For XX and YY G-CW-complexes, the following are bijections of hom-sets:

Ho GSpaces)(X,E˜[N]Y) Ho GSpaces(X NX,E˜[N]) Hom Ho(GSpaces)(X N,E˜[N]Y) = Ho GSpaces(X N,Y) Ho GSpaces(X N,(S 0E˜[N])Y) Ho GSpaces(X N,E˜[N]Y) = Ho GSpaces(X N,Y N) \array{ 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) }

(LMS 86, prop. II 9.3 with remark below the proof)

Corollary

On hom-sets of G-spaces Ho GSpaces(X,Y)Ho_{G Spaces}(X,Y), postcomposing with the smashing (S 0E˜[N])Y(S^0 \to \tilde E \mathcal{F}[N]) \wedge Y is isomorphic to restricting along X NXX^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 :

Ho GSpaces(X,Y) Ho GSpaces(X NN,Y) Ho GSpaces(X N,Y) Ho GSpaces(X N,Y N) Ho GSpaces(X,(S 0E˜[N])Y) Ho GSpaces(X N,(S 0E˜[N])Y) Ho GSpaces(X,E˜[N]Y) Ho GSpaces(X NX,E˜[N]Y) Ho GSpaces(X N,E˜[N]Y) \array{ 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) }
Lemma

(geometric fixed point spectra in terms of homotopy fix point spectra)

The partial geometric fixed point functor (Prop. )

Φ N:GSpectraG/NSpectra \Phi^N \;\colon\; G Spectra \longrightarrow G/N Spectra

is given on equivariant suspension spectra Σ G X\Sigma^\infty_G X equivalently by first smashing with E˜[N]\tilde E \mathcal{F}[N] (Def. ) followed by forming the partial plain fixed point spectrum:

Φ NΣ G X(E˜[N]Σ G X) N. \Phi^N \Sigma^\infty_G X \;\simeq\; \big( \tilde E \mathcal{F}[N] \;\wedge\; \Sigma^\infty_G X \big)^N \,.

(Lewis-May-Steinberger 86, Cor. 9.9)

We will also need this here:

Lemma

For XX a G-CW-complex EE a G- CW-spectrum and NGN \subset G a normal subgroup, the partial NN-fixed point spectrum functor on spectra and the plain fixed point functor on spaces are compatible with smash product:

(E˜[N]EX) N(E˜[N]E)X N \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

(Lewis-May-Steinberger 86, prop. II 9.12)

Via inversion of Euler classes

We discuss an explicit formula (Prop. below, due to Lewis-May-Steinberger 86) that expresses equivariant cohomology groups with coefficients in partial geometric fixed point spectra (Prop. ) as the equivariant cohomology groups with coefficients in the original spectrum, but with certain “Euler classes inverted”.

As an application, we show (Example below) that the equivariant stable cohomotopy of the point in certain non-trivial RO(G)-degrees VV surjects onto the corresponding partially equivariant stable cohomotopy in degree 0 (the latter being well-understood: given by the Burnside ring, by this Prop).

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A key role in this discussion is played by those RO(G)-degrees which trivialize when passing to partial fixed points:

Definition

(RO(G)-degrees without non-trivial HH-fixed points)

For HGH \subset G a subgroup, say that a GG-representation VV has no non-trivial HH-fixed points if the fixed point space of VV with respect to the HH-action is the origin (which is necessarily fixed), hence is the zero-representation:

V H=0 V^H \;=\; 0

We also use the following notation, following Lewis-May-Steinberger 86:

Definition

(base change along normal subgroup-inclusions of equivariance-groups)

Given a normal subgroup-inclusion

NAAGAϵAG/N N \overset{\phantom{AA}}{\hookrightarrow} G \overset{ \phantom{A} \epsilon \phantom{A} }{\longrightarrow} G/N

with induced projection ϵ\epsilon to the quotient group G/NG/N this induces various base change adjunctions (on homotopy categories, say), such as on topological G-spaces, to be denoted

(2)G/NSpacesϵ GSpaces G/N Spaces \overset{ \epsilon^\sharp }{\longrightarrow} G Spaces

and on GG-representations, to be denoted

(3)G/NRepϵ *GRep G/N Rep { \overset{ \epsilon^\ast }{\longrightarrow} } G Rep

and on G-spectra, to be denoted

(4)G/NSpectraAAAA() Nϵ GSpectra G/N Spectra \underoverset { \underset{(-)^N}{\longleftarrow} } { \overset{ \epsilon^\sharp }{\longrightarrow} } {\phantom{AA}\bot\phantom{AA}} G Spectra

where the right adjoint () N(-)^N is the partial fixed point spectrum-functor (in contrast to the geometric fixed point functor).

(e.g. Lewis-May-Steinberger 86, above theorem 9.5)

Proposition

(partial geometric fixed point cohomology via inversion of Euler classes)

Let EGSpectraE \;\in\; G Spectra be a G-spectrum with partial geometric fixed point spectrum Φ NEG/NSpectra\Phi^N E \;\in\; G/N Spectra (Prop. ) and let XG/NSpectra finϵ GSpectra X \;\in\; G/N Spectra^{fin} \overset{\epsilon^\sharp}{\longrightarrow} G Spectra be finite GG-CW-spectrum.

Then the G/NG/N-equivariant cohomology groups in RO(G/N)-degree α\alpha of XX with coefficients in the partial geometric fixed point spectrum Φ NE\Phi^N E are equivalently the colimit over the GG-equivariant cohomology groups of ϵ X\epsilon^\sharp X (2) with coefficients in EE, but in RO(G)-degree ϵ *α\epsilon^\ast \alpha (3) plus a shift by all those representations VV that have no nontrivial NN-fixed points (Def. ):

(5)(Φ NE) G/N (X)lim{V|V N=0}E G ϵ *+V(ϵ X), (\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) \,,

where the colimit is over the diagram that has precisely one morphism for every inclusion V 1V 2V_1 \subset V_2 of GG-representations without non-trivial NN-fixed points (Def. )

E G ϵ *+V 1(ϵ X)()χ V 2V 1E G ϵ *+V 2(ϵ X) 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)

given by smash product with the Euler class

χ V1E G V(S V)(S 0S V) *E G V(S 0) \chi_{V} \;\coloneqq\; 1 \in E_G^V(S^V) \overset{ (S^0 \to S^V)^\ast }{\longrightarrow} E_G^V(S^0)

of VV 2V 1V \coloneqq V_2 - V_1.

(Lewis-May-Steinberger 86, chapter II, prop. 9.13)

Proposition

(comparison map to partial geometric fixed point cohomology)

Prop. provides a canonical comparison morphism, to be denoted

(6)E G ϵ *(ϵ X)AAp E N(X)AA(Φ NE) G/N (X) 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)

from the GG-equivariant cohomology groups with coefficients in EE to those with coefficients in the partial geometric fixed point spectrum: Namely the component of the colimiting cocone(5) at stage V=0V = 0:

E ϵ *(ϵ X) ()χ V E ϵ *+V(ϵ X) p E N(X) (Φ NE) G/N (X) \array{ 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) }

This component is equal to the following composite of isomorphisms with [N]\mathcal{F}[N]'-localization L [N]L_{\mathcal{F}[N]'} (Def. ):

(7)p E N(X):E ϵ *α(ϵ X) =Ho GSpectra(ϵ Σ G/N X,Σ G S ϵαE) Ho GSpectra(ϵ Σ G/N X,Σ G S ϵαS 0E) L [N]Ho GSpectra(ϵ Σ G/N X,Σ G S ϵαE˜[N]E) Ho G/NSpectra(Σ G/N X,(Σ G S ϵ *αE˜[N]E) N) Ho G/NSpectra(Σ G/N X,(S ϵ *α) N(E˜[N]E) N) Ho G/NSpectra(Σ G/N X,S αΦ NE) =(Φ NE) α(X) \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}
Proof

This follows from the proof of (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 with S 0E˜S^0 \to \tilde E \mathcal{F}, up to re-identifications:

  1. The first equality in (7) is the definition of cohomology classes;

  2. the second step is the unitor isomorphism for the tensor unit being the sphere spectrum;

  3. the third step is smashing with S 0E˜[N]S^0 \to \tilde E \mathcal{F}[N], which is \mathcal{F}'-localization by Prop. ;

  4. the fourth step is the hom-isomorphism for the adjunction (ϵ () N)( \epsilon^\sharp \dashv (-)^N ) from (4);

  5. the fifth step is application of Lemma ;

  6. the sixth step is the evident identification (S ϵ *α) N=S α(S^{\epsilon^\ast \alpha})^N = S^\alpha in the first smash factor, and is Lemma in the second factor.

  7. the seventh step is again the definition of cohomology.

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Examples

Example

(equivariant stable cohomotopy of the point in non-trivial RO(G)-degree)

Let GG be a finite group. Then the canonical comparison morphism (6) from Def. exhibits the GG-equivariant stable cohomotopy group of the point in any RO(G)-degree VV that has trivial NN-fixed points (V N=0V^N = 0, Def. ) as a group extension of the G/NG/N-equivariant stable cohomotopy of the point in RO(G/N)-degree zero, hence of the group underlying the Burnside ring A(G/N)A(G/N) (this Prop.):

(8)𝕊 G V(*)AepiA𝕊 G/N 0(*)A(G/N). \mathbb{S}_G^{V}(\ast) \overset { \phantom{A} \text{epi} \phantom{A} } {\longrightarrow} \mathbb{S}_{G/N}^0(\ast) \simeq A(G/N) \,.
Proof

First observe that, in the given situation, the comparison morphism p 𝕊 N(*)p_{\mathbb{S}}^N(\ast) (6) is indeed of the form shown, up to isomorphism: We are in the situation of Prop. for

  1. XΣ G/N (* +)=Σ G/N S 0X \coloneqq \Sigma^\infty_{G/N}(\ast_+) = \Sigma^\infty_{G/N} S^0, which is clearly a finite G/NG/N-CW-spectrum;

  2. EΣ G V𝕊 GΣ G S VE \coloneqq \Sigma^V_G\mathbb{S}_G \coloneqq \Sigma^\infty_G S^V the VV-shifted GG-equivariant sphere spectrum, being the G-spectrum representing GG-equivariant stable cohomotopy, by definition;

  3. Φ NEΣ G/N (S V) NΣ G/N S 0𝕊 G/N\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/NG/N-equivariant sphere spectrum, by (1) and by assumption on VV.

Hence with all identifications made explicit, the morphism (8) in question is the composite

(9)𝕊 G V(*)(Σ G S V) G 0(*)AAp Σ G V𝕊 G N(*)AA(Φ NΣ G S V) G/N 0(*)(Σ G/N S 0) G/N 0(*)𝕊 G/N 0(*)A(G/N) \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)

of p Σ G V𝕊 G N(*)p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) with a sequence of isomorphisms, and hence our task is to prove that p Σ G V𝕊 G N(*)p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) is a surjection.

We first prove this for the case that V=0V = 0. In this case the identification with the Burnside ring (via this Prop.) applies also to the domain cohomology group:

𝕊 G V(*)limVGRepHo GSpaces(S V,S V)A(G), \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) \,,

By Prop. the comparison morphism acts on this by smashing the codomain of the hom-sets with (S 0E˜[N])(S^0 \to \tilde E \mathcal{F}[N]). But by Corollary this is equivalent to restricting to NN-fixed point spaces so that (8) becomes simply the projection of Burnside rings

A(G) limVGRep Ho GSpaces(S V,S V) 𝕊 G 0(*) () N limVGRepHo GSpaces(S V,S V(S 0E˜[E])) = limVGRepHo GSpaces((S V) NS V,S V) p 𝕊 N(*) A(G/N) limWG/NRep Ho G/NSpaces(S W,S W) 𝕊 G/N 0(*) \array{ 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{ \array{ \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) }

sending any G-set KK to its subset K NK^N of NN-fixed points regarded with its residual G/NG/N-action.

This is clearly surjective. (The irreducible elements on the right are the isomorphism classes of the transitive G/NG/N-actions (G/N)/H(G/N)/H for HG/HH \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. says that the comparison map for V=0V = 0 is one coprojection map of a colimiting cocone-diagram, which for each GG-representation VV without non-trivial NN-fixed points (Def. ) contains a cocone component of the following form:

(10)A(G) = (Σ G S 0) G 0(*) ()χ V (Σ G S V) G 0(*) () N epi p 0 p V A(G/N) = (Σ G/N S 0) G/N 0(*) \array{ 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) }

Since we know, as just argued, that the map () N(-)^N on the left is a surjection, the commutativity of this diagram implies that also the component projection p Vp_V is surjective. (Every element cA(G/N)c \in A(G/N) has a lift to c^A(G)\widehat c \in A(G), but then the commutativity of the triangle means that c^χ V\widehat c \wedge \chi_V is a pre-image of cc under p Vp_V.)

This we may use to deduce the statement for the general case, where the codomain of (9) is in degree VV:

By the assumption that the RO(G)-degree VV has no non-trivial NN-fixed points, Prop. says that the colimiting cocone in which the map p Σ G V𝕊 G N(*)p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) appears, by Def. , looks just like the one above, except that it “starts” not in degree 0, but in degree VV:

(11) (Σ G S V) G 0(*) p V (Σ G S 0) G/N 0(*) \array{ && (\Sigma^\infty_G S^V)^0_{G}(\ast) &\to& \cdots \\ & \swarrow_{\mathrlap{p_V}} \\ (\Sigma^\infty_G S^0)^0_{G/N}(\ast) }

In particular the cocone in (10) restricts to a cocone over this sub-diagram in (11), so that the universal property of the cocone in (11) implies an endomorphism ϕ\phi of the abelian group underlying the Burnside ring (Σ G S 0) G/N 0(*)=A(G/n)(\Sigma^\infty_G S^0)^0_{G/N}(\ast) = A(G/n) such that

(12)p V=ϕp Σ G V𝕊 G N(*). p_V \;=\; \phi \;\circ\; p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) \,.

Since p Vp_V is surjective, it is now sufficient to prove that this ϕ\phi is in fact an isomorphism.

To see this, observe that, since GG is a finite group by assumption, the abelian group underlying the Burnside ring A(G/N)A(G/N) is a finitely generated free abelian group (spanned by the cosets (G/N)/H(G/N)/H as HH ranges over the finite set of conjugacy classes of subgroups of G/NG/N ). By the structure theory of free abelian groups, this means that ϕ\phi may be represented by a matrix in Smith normal form. Specifically, since ϕ\phi is an endomorphism, it is represented by a square matrix in Smith normal form. Since ϕp Σ G V𝕊 G N(*)\phi \circ p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) is surjective, by (12) and the surjectivity of p Vp_V established before, this implies that ϕ\phi is represented by a diagonal matrix all whose diagonal entries are non-vanishing and invertible, hence that ϕ\phi is in fact an isomorphism.

With this, (12) says that with p Vp_V also p Σ G V𝕊 G N(*)p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast) is surjective.

References

Relation to spectral Mackey functors:

Last revised on December 22, 2018 at 12:53:11. See the history of this page for a list of all contributions to it.