$RO(G)$-graded homotopy groups vs. Mackey functors

Question

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.

Also, I've looked through other similar MO questions, but I didn't find answers to the questions below. Of course, please let me know if I missed anything!

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In motivic homotopy theory, the category of motivic spectra comes with a set of "bigraded spheres", $S^{i,j} = \Sigma^i (\mathbb{G}_m)^{\wedge j}$. It is well-known that homotopy classes of maps out of these don't detect equivalences. This gives rise to a notion of "cellular" motivic spectra (as studied by Dugger--Isaksen): these sit as a right localization -- that is, a full subcategory whose inclusion admits a right adjoint -- which is by definition the largest subcategory in which bigraded homotopy groups do detect equivalences.

Back in equivariant homotopy theory, the analogous "tautological Picard elements" are the virtual representation spheres $S^V$, i.e. $V$ is a formal difference of finite-dimensional $G$-representations. These corepresent what are therefore known as "$RO(G)$-graded homotopy groups".

A. If a map $X \to Y$ of $G$-spectra induces an isomorphism $[S^V,X] \xrightarrow{\cong} [S^V,Y]$ for all $V \in RO(G)$, is it necessarily an equivalence?

I would of course be interested to hear partial answers, e.g. if this is only known when $G$ is finite/discrete but is open for $G$ a (compact) Lie group -- similarly if this depends on what $G$-universe I'm working over (though the answer is pretty obviously "no" e.g. for the trivial universe).

Moreover, if the answer to the above is "no", then I have a follow-up question.

A'. What is known about the subcategory of cellular $G$-spectra? Is it known to contain or not contain any particular $G$-spectra of interest?

Another question I have is the following.

B. Are there any particular $G$-spectra of interest with known $RO(G)$-graded homotopy groups?

In my limited understanding, the more common notion of "homotopy groups" in the equivariant world are given by homotopy classes of maps out of the stabilized orbits $\Sigma^n \Sigma^\infty_+ (G/H)$ for $n \in \mathbb{Z}$, which altogether assemble into a Mackey functor. By definition, these detect equivalences. (The motivic analog would be mapping out of $\Sigma^{\infty+n} X_+$ for all $X$ in the Nisnevich site.)

C. Besides the obvious ones, are there any known relationships between $RO(G)$-graded homotopy groups and the "homotopy" Mackey functor?

Answer 1

I can answer your first question in some special cases.

Let $p$ be a prime and $G=C_p$ the cyclic group of order $p$. If $p=2$, the answer to your question is yes and if $p$ is odd, then it is no.

First let me rephrase the question. Fix $f\colon X\to Y$ and define a full subcategory $\mathcal{C}_f \subset \mathrm{Sp}_G$ of $G$-spectra as follows: A $G$-spectrum $A$ is in $\mathcal{C}_f$ precisely when $$f_*\colon [\Sigma^n A,X]\to [\Sigma^n A,Y]$$ is an isomorphism for all $n\in \mathbb{Z}$. Note that $\mathcal{C}_f$ is a localizing subcategory (i.e., it is a thick subcategory and closed under arbitrary coproducts).

Let $\mathrm{Lin}\subset \mathrm{Pic}(\mathrm{Sp}_G)$ be the linear representation spheres (i.e, the $G$-spectra of the form $S^{V-W}\simeq S^V\wedge DS^W$ where $V$ and $W$ are real $G$-representations). We can now rephrase your question in either of the following forms:

1. Does the following implication hold for all $f$: $\mathrm{Lin}\subset \mathcal{C}_f\implies {C_p}_+\in \mathcal{C}_f$?
2. Is $\mathrm{Sp}_G$ the smallest localizing subcategory containing $\mathrm{Lin}$?

Indeed by definition, a map $f\colon X\to Y $ is a $G$-equivalence, if it induces an isomorphism $$f_*\colon [\Sigma^* G/H_+, X]\to [\Sigma^* G/H_+, Y],$$ for every (closed) subgroup $H\subset G$. When $G=C_p$, the only subgroups are the trivial subgroup and the whole group $G$. Under the hypothesis of 1., the map $f_*$ is an isomorphism by assumption when $H=G$, so we just need to show $G/e_+={C_p}_+\in \mathcal{C}_f$. This directly relates to condition 2.: The smallest localizing category containing $S^0\in \mathrm{Lin}$ and ${C_p}_+$ is $\mathrm{Sp}_{C_p}$.

When $p=2$ and $\sigma$ is the real sign representation, then the cofiber sequence: $$ {C_2}_+ \to S^0\to S^{\sigma} $$ shows 1. holds.

Suppose that $p$ is odd and 2. holds, we will derive a contradiction. Let $g\colon Z\to {C_p}_+$ be a $\mathrm{Lin}$-cellularization of ${C_p}_+$. In other words, it is a colocalization with respect to $\mathrm{Lin}$; $Z$ is constructed by iteratively gluing 'cells' from $\mathrm{Lin}$ (just as in a CW-approximation) and $g$ induces an isomorphism on $RO(G)$-graded homotopy groups. Since we are assuming 2., $g$ is actually a $G$-equivalence.

Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or, alternatively, the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a shifted copy of $\mathbb{Q}$ with a trivial action). Now we have a cofiber sequence $$\mathbb{Q}[C_p]^{C_p}\to \mathbb{Q}[C_p]\to \mathbb{Q}[C_p]/(\mathbb{Q}[C_p]^{C_p}).$$ Now $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by trivial representations if and only if the cofiber is. However the only map from a trivial rep to the non-trivial cofiber is 0 (note this would fail in characteristic $p$), so the cofiber fails to be in the localizing subcategory generated by the trivial reps and we obtain a contradiction.

If instead of working over the sphere, we work in the category of $KU_G$-modules, then I can say a bit more. Let $G$ be a connected compact Lie group with $\pi_1 G$ torsion-free. Then it is a result of Akhil Mathew, Niko Naumann, and myself that a map $f\colon X\rightarrow Y$ of $KU_G$-modules is an equivalence precisely when it induces isomorphisms: $$ f_*\colon [\Sigma^* S^0, X]\to [\Sigma^* S^0, Y].$$ So here, you don't even need all of the representation spheres, just the trivial representations. This result follows from Thm. 8.3 of Nilpotence and descent in equivariant stable homotopy theory.

I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading (which is what you get when you quotient $RO(G)$ by the equivalence relation $V\sim W\iff S^V\simeq S^W$), I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane spectrum associated to a Mackey functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.

Answer 2

A. The brackets are the same computed in any model, as you say, and for most that entails fibrant approximation. For genuine $G$-spectra (complete universe), $G$ a compact Lie group, it goes back to LMS http://www.math.uchicago.edu/~may/BOOKS/equi.pdf that a map is a weak $G$-equivalence if it induces an isomorphism on $Z$--graded homotopy groups, so on the $\pi_n^H$ for all integers $n$ and (closed) subgroups $H$. That is the right definition of weak $G$-equivalence by Theorem I.4.6 there, which says that a map is a weak $G$-equivalence if and only if it is a spacewise weak $G$-equivalence. Taking $H=G$, one is looking at trivial representations, and so the question amounts to asking when looking at all spheres $S^V$ for $V\in RO(G)$ is equivalent to looking at all $G/H_+\wedge S^n$. Justin's answer says that this is true for $G = Z/2$ and false for $G=Z/p$ for an odd prime $p$.

A'. For G-CW spectra, as defined in LMS where all $G$-spectra are fibrant, a map is a weak $G$-equivalence if and only if it is a $G$-homotopy equivalence. For other models, one must first fibrant approximate and then construct $G$-CW spectra, giving the right model independent notion. Such $G$-CW spectra are not generally well related to model theoretic cell $G$-spectra. See Chapter 24 of http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf for discussion.

B. Very few, as Justin says. Don't have time to think this through right now.

C. Question is cryptic. Spell it out and I can try to answer. Not clear what you have in mind.