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I'm sure this must be well known, but I could not find any references.

My basic question is: Are there "abstract" descriptions of the geometric fixed point functors in equivariant stable homotopy theory, not tied to the Lewis-May construction of the stable quivariant homotopy category using a complete universe?

Of course this is rather vague. I will give a bit of background, and then formulate some more concrete questions at the end.

Background

Let $G$ be a finite group. I denote the stable equivariant homotopy category with respect to $G$ by $SH(G).$ Without getting into too many details, it relates to the category of spaces with $G$-action in a similar way that the ordinary stable homotopy category $SH$ relates to ordinary spaces. It is a symmetric monoidal, triangulated category. One of its most important features is that all the representation spheres of $G$ are invertible, not just the ordinary ones. Of course, $SH(G)$ by itself is "not good enough", we also need point set level models for it, and there are several.

For any subgroup $H$ of $G,$ there exists a functor $\Phi^H: SH(G) \to SH,$ called the "geometric fixed points functor". These functors have the following properties (not an exhaustive list I'm sure):

  1. Monoidality: the functors $\Phi^H$ are monoidal (and triangulated).
  2. Compatiblity with space-level fixed points: if $X$ is a $G$-space and $\Sigma^\infty_G$ denotes the suspension spectrum, then we have $\Sigma^\infty X^H \simeq \Phi^H \Sigma^\infty_G X$ (where $X^H$ is the subspace of fixed points, viewed as an ordinary space, and $\Sigma^\infty$ denotes the ordinary suspension spectrum).
  3. Whitehead theorem: a map $f: E \to F$ in $SH(G)$ is a weak equivalence if and only if all of the $\Phi^H(f)$ are.

As I said before, the only construction of these functors I could find is using a specific model for $SH(G),$ and it is not clear to me how to transfer this to others.

Here are some concrete questions:

  1. The space-level fixed point functor is given by $X^H = S(G/H, X),$ where $S$ denotes the enrichement of $G$-spaces in ordinary spaces (i.e. if $FX$ denotes the $G$-space $X$ with $G$-action forgotten, then $S(X, Y)$ is the subspace of $S(FX, FY)$ consisting of equivariant maps, where $S(FX, FY)$ is the usual mapping space). As it so happens, $SH(G)$ is a spectral category (provided with mapping spectra), let $\mathcal{S}$ denote the enrichment. Then the most analogous definition would be $\Phi'^H(E) = \mathcal{S}(\Sigma^\infty_G G/H_+, E),$ but I suspect this is not the same as $\Phi^H.$ Indeed it satisfies (2) andedit: that's not actually true (3) but I do not see why it should satisfy (1). However, this is the kind of description I would hope for.

  2. A different description of $SH(G)$ is in terms of symmetric spectra on $G$-spaces, with the regular representation being inverted (I believe). How can the geometric fixed point functors be described in this language?

  3. Yet another description is as follows: let $\mathcal{O}$ denote the full spectral subcategory of $SH(G)$ with objects $\Sigma^\infty_G G/H_+$ (this is slightly sloppy). One can consider the category of "spectral presheaves on $\mathcal{O}$", $\mathbf{Pre}(\mathcal{O}, SH)$ (this notation is even sloppier). As it turns out, this category is equivalent to $SH(G)$. Again, how does the geometric fixed points functor look like in this framework? (Of course this question isn't really well-posed, since the answer depends on how one describes $\mathcal{O}$ and $SH$ in the first place.)

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    $\begingroup$ I don't know if this is right, so I'll leave it as a comment. But my guess is that it is characterized by the properties: (i) it preserves homotopy colimits, and (ii) it makes the diagram commute between G-spaces, spaces, G-spectra, and spectra- i.e. geometric fixed points of a suspension spectrum are suspension spectra of geometric fixed points $\endgroup$ – Dylan Wilson Apr 24 '14 at 14:08
  • $\begingroup$ Ah, I may have to add monoidality... I haven't thought about it. $\endgroup$ – Dylan Wilson Apr 24 '14 at 14:13
  • $\begingroup$ Well I think geometric fixed points is a left adjoint, so preserves homotopy colimits, and your (ii) determines it on cells, so (i) and (ii) determine it on all spectra, no? But then monoidality is a bit of a mystery (to me). $\endgroup$ – Tom Bachmann Apr 24 '14 at 14:18
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    $\begingroup$ Well I guess you also need to know it deloops well. Then you'd get monoidality because smashing commutes with hocolims and everything is a hocolim of (deloopings of) suspension spectra. $\endgroup$ – Dylan Wilson Apr 24 '14 at 14:27
  • $\begingroup$ (and monoidality for those follows from the same claim for spaces) $\endgroup$ – Dylan Wilson Apr 24 '14 at 14:28
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Specifically addressing question (2), the category of equivariant symmetric spectra for $G$ with respect to the sphere $S^{\rho_G}$ for the regular representation is put together systematically in Mandell's paper "Equivariant symmetric spectra," Contemporary Mathematics, vol 346 (a preprint is available on his webpage.) In section 9.5 of the preprint version he discusses the geometric fixed point functor.

It is pretty straightforward in this framework. If your object $X$ consists of a sequence of $G \times \Sigma_n$-spaces $X_n$ with structure maps $S^{\rho_G} \wedge X_n \to X_{1+n}$ (with appropriate equivariance conditions), one model for the $H$-geometric fixed points is as follows. Noting that we have a fixed-point identification $(S^{\rho_G})^H = S^{\rho_H}$, we take $(\Phi^H X)_n = (X_n)^H$, with structure maps $$ S^{\rho_H} \wedge (X_n)^H = (S^{\rho_G} \wedge X_n)^H \to (X_{1+n})^H. $$ This makes it a very literal fixed-point application in this framework, and it is very different than the function spectrum $F\left(\Sigma^\infty_G G/H_+, X\right)$.

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  • $\begingroup$ I habe no idea how i missed this, i'm positive i typed "equivariant symmetric spectra" into Google more than once. I'll check your reference in detail on Monday. $\endgroup$ – Tom Bachmann Apr 26 '14 at 6:11
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The geometric fixed point construction can easily be described in terms of constructions present in any good model for the equivariant stable category. For notational simplicity, I'll restrict to the case $H = G$. In general, one can first apply the change of groups functor from $G$-spectra to $H$-spectra and then apply the geometric $H$-fixed point functor. Let $\mathcal P$ be the family of proper subgroups of $G$. There is a classifying $G$-space $E\mathcal P$, and we let $\tilde E \mathcal{P}$ denote the cofiber of the evident based $G$-map $E\mathcal{P}_+ \to S^0$. Any good model is tensored over $G$-spaces, so for any $G$-spectrum $X$, we have the $G$-spectrum $X\wedge \tilde E \mathcal{P}$. Its categorical $G$-fixed point spectrum is the geometric fixed point spectrum of $X$. (This is pointed out briefly in Section XVI.3 of Equivariant Homotopy and Cohomology Theory. I've ignored model theoretic niceties for clarity.)

Edit: In the context of orthogonal $G$-spectra, Mandell and I gave a reasonably comprehensive treatment in Section 4 of Chapter V of ``Equivariant orthogonal $G$-spectra and $S$-modules'', all in a more general framework of normal subgroups and quotient groups. We give several definitions and comparisons (see Lemma 4.15 and Proposition 4.17, including the context-free one I described above). The functor preserves cofibrations and acyclic cofibrations by Proposition 4.5, but I see no reason to think it is a left adjoint: I described it in the first place as a composite of a left and a right adjoint. The monoidal property is Proposition 4.7, and the compatibility with space-level fixed points is Corollary 4.6. The Whitehead theorem is not there, but is a comparison with the usual criterion in terms of "categorical" fixed points (aka Lewis-May fixed points) by the implicit isotropy separation cofibration and induction. The term "categorical" refers to the fact that this fixed point functor is right adjoint to the functor from nonequivariant orthogonal spectra, viewed as $G$-trivial naive $G$-spectra, to genuine $G$-spectra obtained by change of universe.
You first change to the trivial universe and then take level wise fixed points.

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  • $\begingroup$ Two questions: (1) what is the "categorical fixed point spectrum"? For e.g. symmetric spectra, do you just take fixed points termwise? And for the presheaf version, do you restrict to [G/G]? (2) Perhaps more importantly, how do the properties I listed follow from this construction? $\endgroup$ – Tom Bachmann Apr 25 '14 at 13:59
  • $\begingroup$ I answered in an edit. $\endgroup$ – Peter May Apr 26 '14 at 0:24
  • $\begingroup$ Thanks a lot, I'll think it through on Monday (currently on mobile). Re adjointness: i was quickly (and carelessly) misreading theorem 7.1 of "fixed point adjunctions for equivariant module spectra" $\endgroup$ – Tom Bachmann Apr 26 '14 at 6:14

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