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A question a little into the blue, please bear with me:
It is natural to consider global equivariant stable homotopy theory for collections of finite subgroups. If we take the collection of finite subgroups of $SO(3)$ (or maybe $SU(2)$) then these are the finite groups in the ADE classification. This is clearly a special case, of sorts.
Is there any dedicated discussion of global homotopy theory in this ADE situation?
For what it’s worth, I am still motivated by trying to find a cohomological home for the gauge enhancement mechanism. Whatever that means, it’s a fact that in M-theory one is considering spaces with actions by precisely the finite groups in the ADE classification, and hence any cohomology theory one would like to associate here would plausibly be global equivariant with respect to the collection of finite groups in the ADE classification.
In particular, I have some indication that the correct cohomology theory to use should be a version of cohomotopy, but I know that plain stable cohomotopy is not correct, because the unstable homotopy group $\pi_7(S^4)$ needs to remain non-torsion. I am wondering if in the global sphere spectrum for the collection of finite ADE groups this could work out, for some suitable representation sphere…
So the collection of finite subgroups of $SO(3)$ is a ’global family’ in the sense of Schwede’s book Global homotopy theory?
Definition 7.1. A global family is a non-empty class of compact Lie groups that is closed under isomorphism, closed subgroups and quotient groups.
There’s no chance of their being a quotient which is not a subgroup, as when the quaternion group of order 8 has the Klein group as quotient but not subgroup?
[Hmm, the quaternion group is a finite subgroup of $SU(2)$. But I guess the Klein group is also a subgroup.]
There is work on “global families of finite groups” as in Symmetric spectra model global homotopy theory of finite groups.
Sorry, I should have been more to the point: there are many different ways known to model G-equivariant stable homotopy groups. Independently of that, what I am after first of all is this:
For which (if any) finite groups $G$ with 3-dimensional orthogonal representation $V_3$ does the quaternionic Hopf fibration represent a non-torsion element in the $G$-equivariant stable homotopy groups of spheres in RO(G)-degree $V_3$?
I have found one computation that goes in this direction, Araki-Iriye 82, summarized here.
They consider the simplest case $G = \mathbb{Z}_2$. And they find in this case that the complex Hopf fibration does become non-torsion. The quaternionic Hopf fibration, however, they show is still of order 24, as in the non-equivariant case.
But maybe if we go beyond just $\mathbb{Z}_2$, also the quaternionic Hopf fibration has a chance to become non-torsion?
Sounds like a question for MathOverflow.
Was the idea from #1 that even if there’s no positive answer for #3 with any particular $G$, that there might be a family for which the global answer is what you want?
The idea of #1 was that I didn’t want to fix one of the ADE finite groups, but consider them all at once. But before I worry about that, I should see if for any fixed one I get what I need in the first place.
Here is a reason to think it could work:
the construction of the complex Hopf fibration as a non-torsion element in the $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres in Araki-Iriye 82, top of p.24 relies on the fact that a) multiplication of complex numbers is $\mathbb{Z}/2$-equivariant with respect to complex conjugation, and b) that the Hopf construction applied to the unit circle in the complex numbers is compatible with this action.
Now, analogously, the product of quaternions is of course equivariant with respect to the $SO(3)$-action on their imaginary part. That suggests that in analogy the quaternionic Hopf fibration should represent elements in the G-equivariant stable homotopy groups of sphere in the RO(G)-degree given by the restriction of the defining representation of $SO(3)$ to the inclusion of finite subgroups, hence to the finite groups in the ADE series.
I’d like to know these equivariant stable homotopy groups in “ADE degrees” in total, but first of all I’d need to know if any one of them contains a non-torsion summand.
I wonder what effect the “in total” has. Does it tend to remove torsion (something like the least common torsion for the different $G$)?
If so, might you not have torsion for all groups, but ’coprime’ so it disappears in total?
Ah, no, the global version would just relate all the $G$-equivariant homotopy groups to each other, for all $G$ in the collection. But that a Hopf fibration goes from being a torsion generator in plain stable homotopy groups to a non-torsion generator in $G$-equivariant stable homotopy groups must be an effect of the $G$-equivariance at any fixed $G$: The $G$-equivariance makes the Hopf fibration more rigid, where without equivariance its square may be “unwound” to become stably trivial, for $G$–equivariance this unwinding has to happen $G$-equivariantly, too, and that may make it impossible. This is, if I see correctly,what does indeed happen with the complex Hopf fibration with $\mathbb{Z}/2$-equivariance. It seems natural to wonder whether it also happens for the quaternionic Hopf fibration with equivariance with respect to any finite subgroup of $SO(3)$.
I have forward the question to MO.
And you have an answer.
Thanks to Charles! I was getting a bit bogged down in unwinding technicalities, but his argument is much more direct and quite simple.
It occurs to me that the exlusion of the cyclic among the finite ADE groups which is found to be necessary thereby actually matches the situation in the application that motivates me here: the same exclusion is seen for instance in
See p. 3.
I started a remark on this at G2-manifold – With ADE orbifold structure but no time right now to expand on it properly.
I am wondering if this algebraic K-theory of A-type singularities could be related to the above story of stable cohomotopy at D- and E-type singularities
The next thing I need to figure out is this: given that finite non-cyclic subgroup of $SO(3)$ and with $S^4$ regarded as a $G$-space, then given an 11-dimensional circle fibration of $G$-spaces, I need to understand the corresponding equivariant Atiyah-Hirzebruch spectral sequence for $G$-equivariant 4-shifted stable cohomotopy $[X_{11},\Sigma^\infty_G S^4]_G$, and I need to check that it gives a kind of correction to the Atiyah-Hirzebruch spectral sequence for self-dual topological K-theory on $X_{10}$.
Is there any discussion of Atiyah-Hirzebruch-type spectral sequences for genuine $G$-equivariant cohomology?
I am preparing some talk notes here: Generalized cohomology of M2/M5-branes (schreiber)
Is there any discussion of Atiyah-Hirzebruch-type spectral sequences for genuine G-equivariant cohomology?
I see that theorem 3.1 in Kronholm 10 goes in this direction. It gives the analog of the AHSS for Mackey functors. Combined with the Greenless-May splitting that gives at least some of the information.
Great to see the role of the E-series groups in fundamental physics as Plato more or less predicted :)
Given that you’re interested in the case that the AHSS is applied to a fibration with fibre $S^1$, does this simplify things somewhat? Can we pin down what sort of actions one gets on the circle by (A)DE subgroups of $SO(3)$?
Given that you’re interested in the case that the AHSS is applied to a fibration with fibre $S^1$, does this simplify things somewhat?
It should, yes.
Can we pin down what sort of actions one gets on the circle by (A)DE subgroups of $SO(3)$?
A scan of some possibilities is Atiyah-Witten 01. I am not sure though if there are any general conclusions to be drawn from this zoo. But to get a feeling for what is being considered, go for instance to “Case III” on p.37. This is about 7-manifolds that are cones over $S^3 \times S^3$ obtained by filling one of the $S^3$ to a 3-ball. The ADE-action then is via finite subgroups of $SU(2) = S^3$ acting on either factor or diagonally, and similarly for the $U(1)$-action. You should browse around a bit more in the article to see the possibilities being considered.
I realize that I have a basic confusion:
is the unit sphere $S^7_{SO(3)} = S(\mathbb{H}\times \mathbb{H})$ (with the $SO(3)$-action coming from the diagonal action of $SO(3) = Aut_{\mathbb{R}}(\mathbb{H})$) $SO(3)$-equivariantly homeomorphic to a representation sphere, $S^V$, hence to a 1-point compactification of a linear and orthogonal $SO(3)$-representation on $\mathbb{R}^7$?
And is it actually strictly necessary to use such representation spheres to have an equivariant suspension isomorphism, or could we also speak about (de-)looping with respect to more general spheres with $G$-action? Or does it not matter?
Maybe spheres with a linear G-action with at least one fixed point? This is how one picks a point to stereographically project from, I suppose. One calls this the point at $\infty$. But perhaps these are exactly equivalent to representation spheres?
But considering the action groupoid associated to this, one can probably find a point with a slice that has such a fixed point, and this slice might itself be an H-sphere, for H the stabiliser of that fixed point…
And I claim (as per my email) the the V you are looking for in #18 is $\mathbb{H}\times Im(\mathbb{H})$, where the $SO(3)$-action is $Aut_\mathbb{R}(\mathbb{H})$ on the first factor and its restriction on the other factor.
with at least one fixed point?
At least two fixed points. 0 and $\infty$ :-)
I claim this follows from being a linear action ;-) But more generally, one probably wants that if linearity is dropped.
And I claim (as per my email) the the V you are looking for in #18 is $\mathbb{H}\times Im(\mathbb{H})$, where the $SO(3)$-action is $Aut_\mathbb{R}(\mathbb{H})$ on the first factor and its restriction on the other factor.
Okay, thanks. But I am being dense. How do you see the equivalence?
Oh, I get it.
I’ve added the general statement about constructing representation spheres using S(R×V) to representation sphere. I’m removing the redirect of stereographic projection to Riemann sphere, and will make a new page soon, since there was nothing about the former on the page for the latter.
Thanks!
Okay, with that out of the way, now back to the equivariant Atiyah-Hirzebruch spectral sequence.
So I suppose what we should be considering is really the cohomology theory
$H^{(\mathbb{R}^{11}-\mathbb{R}^4_G - \mathbb{R}^7_G)+ \bullet}(-, \Sigma^\infty_G S^4_G) \,,$where I am writing $\mathbb{R}^4_G$ for the ($G \hookrightarrow SO(3)$)-representation on $\mathbb{R} \oplus Im(\mathbb{H})$ and $\mathbb{R}^{7}_G$ similarly for the representation on $\mathbb{H} \oplus Im(\mathbb{H})$, in accord with what David Roberts has been pointing out above.
This is of the form that
the spectral sequence from theorem 3.1 in Kronholm 10 applies; and
its $E_2$-page for an $S^1$-fibration gives non-torsion contributions in the degrees in which $H^\bullet(-, \Sigma^4 H \mathbb{Z} \oplus \Sigma^7 H \mathbb{Z})$ does, too (plus possibly loads of further torsion contributions that I have no idea about what they will be)
So this gives, I think, that rationally the cohomology theory called $H^{(\mathbb{R}^{11}-\mathbb{R}^4_G - \mathbb{R}^7_G)+ \bullet}(-, \Sigma^\infty_G S^4_G)$ satisfies the two consistency conditions, that it looks like degree-4 cohomotopy and reduces on 11-dimensional circle fibrations to twisted K-theory on the 10d base, rationally.
The big question then is to which extent it actually coincides with twisted K-theory on the base non-rationally. This looks like it should be interesting either way, both to the extent that it agrees, as well as to the extent that it differs…
re #17: maybe I was missing the forest for the trees:
let’s focus on the dihedral groups, the D-type finite groups in the ADE-classification. They are generated, as subgroups of $SO(3)$, from a cyclic group $C_p$ and a reflection at the plane of rotation. We should not really fix any one of these but really consider the global equivariant structure as $p$ ranges over the the natural numbers (whence the title of this thread!)
Now, I suppose we are to regard the global dihedral equivariant sphere spectrum as being in particular the global cyclic equivariant sphere spectrum, and as such as a cyclotomic spectrum, which means that we really think of it as a circle-group equivariant spectrum with all the $C_p$ being the finite subgroups of the $S^1$-action.
And then I suppose as we evaluate this now on an 11-dimensional $G$-manifold, it is that very circle action which is the rotation of the M-theory circle fibers.
So it’s not an ADE action and an $S^1$-action, but the $S^1$-action is already subsumed by/reflected in the ADE equivariance, I suppose.
And then it all comes out as it should be: the fixed points are the O-places at the points where the M-theory circle fiber degenerates, hence precisely the O6-planes as in the seminal account in [Sen 97b(https://ncatlab.org/nlab/show/F-theory#Sen97b).
Hm, if it’s really the cyclotomic sphere that appears in the story, then the interpretation must be like so: the cyclotomic structure is the structure of free loop spaces. In the context of M-theory reductions from 11d spacetimes $X\times S^1$ to 10d spacetimes $X$, these appear in the double dimensional reduction, wich sends cocycles $\nabla \colon X \times S^1 \to E$ to
$X \longrightarrow [S^1, X \times S^1] \stackrel{[S^1, \nabla]}{\longrightarrow} [S^1, E] \,.$Hm…
The so-called ’caloron correspondence’ studied by various people from Adelaide may well be appropriate here. Ray Vozzo is author on most of those papers.
Thanks for the reminder. I have now added cross-links between caloron correspondence and double dimensional reduction.
’Caloron’, something to do with heat? Supposedly from here, but must dash.
’Caloron’, something to do with heat? Supposedly from here, but must dash.
I added the recent preprint on ADE G2 orbifolds to G2 manifold and ADE singularity.
@David C
Michael Murray pointed me to this paper, which was actually referred to on Wikipedia. They consider Yang-Mills instantons as being a zero-temperature (or possibly infinitesimal-temperature) state, and the positive finite temperature ’Yang-Mills gas’ solutions they call calorons.
Thanks. Let’s record that at caloron.
Expanded caloron a bit.
Just for the record: I see that there is a classical article Cook-Crabb 93 which discusses H-spaces in parameterized homotopy theory and finds that for spherical fibrations these come precisely from the $\mathbb{A}$-Hopf fibrations ($\mathbb{A} =$ complex, quaternionic, octonionic, they don’t bother to mention the trivial real case) equipped with their $Aut(\mathbb{A})$-equivariant structure.
That is of course closely related to what we have been discussing here. Though I am not sure if it sheds any further light on what we are after here beyond what we already discussed.
also Iriye 95
What is known about the equivariant complex cobordism ring in degree 3, $\pi_3(M U_G)$ ?
(For $G$ some finite group, preferably a subgroup of $SO(3)$.)
I am wondering because, by the (equivariant) ANSS, we know that to compute M2-brane charge in equivariant stable cohomotopy
$X^4 \times S^7 \simeq S^7 \longrightarrow \Sigma^4 \mathbb{S}_G$we may equivalently compute it in equivariant complex cobordism
$S^7 \longrightarrow \Sigma^4 \mathbb{S}_G \longrightarrow \Sigma^4 M U_G$if we furthermore keep the information of the further induced maps into the cosimplicial $M U_G^{\wedge^\bullet}$.
Now a cocycle $S^7 \to \Sigma^4 M U_G$ is of course an element in $\pi_3(M U_G)$ (and there is an equivariant Thom theorem saying that this is indeed still isomorphic to the equivariant complex cobordism ring in degree 3).
This looks interesting with respect to the physics interpretation, because of course non-equivariantly then this vanishes. So we would get here M2-brane charges that vanish in general, except at orbifold fixed points where they correspond to classes of 3-manifolds.
That fits well the physics story of “gauge enhancement” where at the orbifold fixed points there are supposed to be hidden collapsed M2-branes that have shrunk away along the blow-up of the singularity. So if one could see those vanishing cycles kind of stored in $\pi_3(M U_G)$, then that would make a whole lot of sense.
It seems not much is known according to William C.Abram, Igor Kriz in The equivariant complex cobordism ring of a finite abelian group
Perhaps surprisingly, the problem of calculating explicitly tom Dieck’s stable equivariant cobordism ring $(MU_G)_\ast$ has remained open for the last 40 years.
Presumably their results give the A series at least.
Thanks. I glanced over their article, but didn’t yet spot any statement that would tell me which elements I would find in $(MU_{C_p})_3$.
Trouble is that I don’t really have time for this, so maybe I just didn’t look closely enough. I really need to be looking into something else. But if you spot anything, let me know.
I guess section 4 is about as close as it gets. For $n=1$, then $(MU_{C_p})_\ast$ is given by a pullback of rings.
Abram’s thesis says
We have given explicit algebraic descriptions of the equivariant complex cobordism ring $MU_G$ and its corresponding equivariant formal group law for $G$ a finite abelian group. The corresponding description of $MU_{Z/p}$ by Kriz [21] was used by Strickland [31] to give generators and relations for $MU_{Z/2}$. The present work should allow Strickland’s computations to be carried over for more general groups $G$.
[31} is Strickland’s Complex cobordism of involutions
Also from Abram’s thesis:
To obtain his result, Strickland makes use of a pullback diagram of Kriz [21] describing $MU_{Z/p}$, to which we will return. A good first step toward a more general result would be a similar diagram for $MU_G$, for more general $G$.
Where [21] is I. Kriz, “The $Z/p$-equivariant complex cobordism ring”, Homotopy invariant algebraic structures, Contemporary Mathematics, American Mathematical Society Publications, Providence, Volume 239, 1999, pp. 217-223.
Theorem II.1 has the diagram from Kriz, and Theorem II.4 (and the preceding discussion) has the generalisation of that diagram to more general abelian groups.
Thanks! That’s useful. In particular Comezana’s result, cited as theorem I.18 in Abram’s thesis: for $G$ abelian then $(M U_G)_\ast$ is still all in even degrees. That means that to get what I am after, elements in degree 3, the A-series once again won’t do it.
(I’ll be: my old friend, Gustavo Comezana. (-: )
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