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added the statement (here) that the geometric fixed point spectrum of an equivariant suspension spectrum constitues exactly the “first” summand in the tom Dieck splitting of the plain fixed point spectrum
added the same remark also in at tom Dieck splitting
Is there a “partial” geometric fixed point construction?
For $N \subset G$ a normal subgroup inclusion, is there a functor
$\Phi^N \;\colon\; G Spectra \longrightarrow G/N Spectra$such that
$\Phi^N \Sigma^\infty_G X \;\simeq\; \Sigma^\infty_{G/N} X$?
And if so, does this participate in a “partial” tom Dieck splitting with the relative homotopy fixed point functor
$F^N \;\colon\; G Spectra \longrightarrow G/N Spectra$such that
$F^N \Sigma^\infty_G X \simeq \Sigma^\infty_{G/N} X \;\vee\; \cdots$?
[ edit: found the answer to the first question: very bottom of p. 40 here ]
I have tried to write out a proof that
the $N$-fixed point comparison map
$(\Sigma^\infty_G S^V)^0_{G}(\ast) \overset{\phantom{AA} p_V \phantom{AA} }{\longrightarrow} (\Phi^N \Sigma^\infty_G S^V)^0_{G/N}(\ast) = (\Sigma^\infty_{G/N} S^0)^0_{G/N}(\ast) = A(G/N)$from the $G$-equivariant stable cohomotopy of the point in RO(G)-degree $V$ to the $G/N$-equivariant stable cohomotopy in degree 0 is surjective (here)
Maybe there is a gap where I am claiming that the triangular diagram is a colimiting cone in two ways (the apex certainly is, but do I know from this that the component maps $p_V$ really coincide?)
But need to run now, will check or else delete later
Yes, of course, it works: one $p_V$ is surjective by the argument given in the entry. But then, by the universal property of the colimit, it factors through the other $p_V$ followeded by an endo map (of finite abelian groups!) For the latter to be surjective it must be an iso.
Yes, of course, it works: one $p_V$ is surjective by the argument given in the entry. But then, by the universal property of the colimit, it factors through the other $p_V$ followeded by an endo map (of finitely generated free abelian groups!) For the latter to be surjective it must be an iso.
okay, I have polished and completed (I think) the proof of that claim (now here):
Let $G$ be a finite group. Then the canonical comparison morphism exhibits the $G$-equivariant stable cohomotopy group of the point in any RO(G)-degree $V$ that has trivial $N$-fixed points ($V^N = 0$) as a group extension of the $G/N$-equivariant stable cohomotopy of the point in RO(G/N)-degree zero:
$\mathbb{S}_G^{V}(\ast) \overset { \phantom{A} \text{epi} \phantom{A} } {\longrightarrow} \mathbb{S}_{G/N}^0(\ast) \simeq A(G/N) \,.$I was instead thinking about arguing to bridge that remaining gap (here) as follows, but not sure yet:
The proof of prop. II 9.13 on Lewis-May-Steinberger shows that the map in question, which I want to see is surjective, is induced by tensoring with the map
$S^0 \longrightarrow \tilde E \mathcal{F}[N] \,.$But earlier around Prop. 9.4, they seem to show that tensoring with this map is what induces the localization of $G$-spectra to $G/N$-spectra.
This would reduce me to arguing that the localization functor is full. Isn’t that the case?
Will have to think more about this. But now I need to go offline for tonight.
added pointer to
I brought in a batch of further lemmas from Lewis-May-Steinberger 86, II.9 in a new subsection now titled “In terms of smashing localization” (here).
(This section is currently bare bones, using notation from LMS without introducing it. )
Then I used this (in this prop.) to make fully explicit the colimiting degree-0 component that is implicit in the proof of LMS II Prop. 9.13, showing that this is, up to equivalence, given by $\mathcal{F}[N]'$-localization:
$\begin{aligned} p_E^N(X) \;\colon\; E^{\epsilon^\ast \alpha}(\epsilon^\sharp X) & = Hom_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge E \right) \\ & \simeq Hom_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge S^0 \wedge E \right) \\ & \overset{ loc }{\longrightarrow} Hom_{G Spectra}\left( \epsilon^\sharp \Sigma^\infty_{G/N} X \;,\; \Sigma^\infty_G S^{\epsilon \alpha} \wedge \tilde E \mathcal{F}[N]\wedge E \right) \\ & \simeq Hom_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; \left( \Sigma^\infty_G S^{\epsilon^\ast \alpha} \wedge \tilde E \mathcal{F}[N] \wedge E \right)^N \right) \\ & \simeq Hom_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; \left( S^{\epsilon^\ast \alpha} \right)^N \wedge \left( \tilde E \mathcal{F}[N] \wedge E \right)^N \right) \\ & \simeq Hom_{G/N Spectra}\left( \Sigma^\infty_{G/N} X \;,\; S^{\alpha} \wedge \Phi^N E \right) \\ & = (\Phi^N E)^{\alpha}(X) \end{aligned}$[ removed ]
brought in yet more lemmas from LMS86 (also added introduction of more of the notation) such as to extract this corollary (here):
On hom-sets of G-spaces $Ho_{G Spaces}(X,Y)$, postcomposing with the 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 this Lemma:
$\array{ Ho_{G Spaces} \big( X, Y \big) & \overset{ Ho_{G Spaces} \big( X^N \hookrightarrow N, Y \big) }{ \longrightarrow } & Ho_{G Spaces}\big( X^N, Y \big) &\simeq& Ho_{G Spaces}\big( X^N, Y^N \big) \\ {}^{ \mathllap{ Ho_{G Spaces}( X, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y ) } } \big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow {}^{ \mathrlap{ Ho_{G Spaces}( X^N, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y ) } } \\ Ho_{G Spaces}\big( X, \tilde E \mathcal{F}[N] \wedge Y \big) & \underoverset {\simeq} { Ho_{G Spaces} \big( X^N \hookrightarrow X, \tilde E \mathcal{F}[N] \wedge Y \big) } {\longrightarrow} & Ho_{G Spaces}\big( X^N, \tilde E \mathcal{F}[N] \wedge Y \big) }$This then I used to finish off the argument at the beginning of my inductive proof (here) of surjectivity of $\mathbb{S}^V_G(\ast) \to \mathbb{S}^0_{G/N}(\ast)$ (whenever $V^N = 0$), for the case $V = 0$:
In this case the identification with the Burnside ring (via this Prop.) applies also to the domain cohomology group:
$\mathbb{S}_G^V(\ast) \;\simeq\; \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big(S^V, S^V \big) \simeq A(G) \,,$By this Prop. 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 this Corollary this is equivalent to restricting to $N$-fixed point spaces so that the comparison map becomes simply the projection of Burnside rings
$\array{ A(G) &\simeq& \underset{\underset{ V \in G Rep }{\longrightarrow}}{\lim} & Ho_{G Spaces}\big( S^V, S^V \big) &\simeq& \mathbb{S}^0_G(\ast) \\ {}^{\mathllap{ (-)^N }}\big\downarrow && \big\downarrow {}^{{}_{ \mathrlap{ \array{ \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big( S^V, S^V \wedge (S^0 \to \tilde E\mathcal{F}[E]) \big) \\ = \\ \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim} Ho_{G Spaces}\big( (S^V)^N \hookrightarrow S^V, S^V \big) } } } } & && \big\downarrow{}^{ \mathrlap{ p_\mathbb{S}^N(\ast) } } \\ A(G/N) &\simeq& \underset{\underset{ W \in G/N Rep }{\longrightarrow}}{\lim} & Ho_{G/N Spaces}\big( S^W, S^W \big) &\simeq& \mathbb{S}^0_{G/N}(\ast) }$sending any G-set $K$ to its subset $K^N$ of $N$-fixed points regarded with its residual $G/N$-action.
This is clearly surjective.
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