[Math] “abstract” description of geometric fixed points functor

at.algebraic-topologyequivariant-homotopyhomotopy-theory

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.)

Best Answer

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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