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added pointer to
Daniel Dugger, Daniel Isaksen, $\mathbb{Z}/2$-equivariant and R-motivic stable stems, Proceedings of the American Mathematical Society 145.8 (2017): 3617-3627 (arXiv:1603.09305)
exposition in
Daniel Dugger, Motivic stable homotopy groups of spheres (pdf)
I’m trying to figure out where in the table the group in this comment on yours fits. I don’t quite understand the description of the bigrading convention, since both the source and the target are smashes of spheres with sign reps, in the case you want.
What happens to the RO(G)-grading idea with the move to global equivariance? Does Peter May’s warning that it is “not the thing most intrinsic to the mathematics” need to be heeded? Presumably something intrinsic is being a stable object in the kind of equivariant $(\infty, 1)$-topos being sought here.
Then what is the ’logic’ of those motivic spectra?
I’m trying to figure out where in the table the group in this comment on yours fits. I don’t quite understand the description of the bigrading convention, since both the source and the target are smashes of spheres with sign reps, in the case you want.
Right, so degrees on the right appear as negatives of degrees on the left.
So the stable class of a map
$S^{ 5_{sgn} } \longrightarrow S^{ 3 + 1_{sgn} }$is an element which in Araki-Iriye’s convention is in $\pi^S_{ 4, -3 }$ with $p = 5 - 1 = 4$ the net dimension of sign reps, and $q = 0 - 3 = -3$ the net dimension of trivial reps.
Hence in the table this is the entry $(p+q,p) = (1,4)$, where we have the equivariant stable homotopy group $(\mathbb{Z}/2)^2$. So at least after stabilization, the answer to my previous question is “No, there is no non-torsion stuff in that degree”.
After I had written this question I realized that I had looked at this stuff in Araki-Iriye long ago already, then forgotten about it. But then I discovered this table by Dugger summarizing their results, which increases the usability by some orders of magnitude.
This way I learned that I had been asking not quite the right question. What I wanted to see is in which $RO(\mathbb{Z}/2)$-degree of total dimension 4 we can see the charges of the MO5/M5-bound states, which turn out to be perfectly captured by the Cohomotopy of MO5-singularities in RO-degree $5_{sgn}$.
To see this also in degree-4 Cohomotopy, subject to the constraint that orientation behaviour is respected, we find from the table now that we can either map $S^{5_{sgn} + 1} \longrightarrow S^{ 3_{sgn} + 1 }$ or $S^{5_{sgn} + 3} \longrightarrow S^{ 1_{sgn} + 3 }$.
(This is an equivariant analog to how the 4-sphere sees M2-brane charge a priori measured by $S^7$, due to the fact that there is a non-torsion element $S^7 \to S^4$. What we see here is that/how equivariantly the 4-sphere sees further brane species this way, here the MO5/M5-bound system.)
This may all sound mysterious. I’ll be trying to write it out comprehensively and cleanly.
But will be busy the next days. Tomorrow flying back from family vacation across continents, then Domenico will be visiting for a week and we’ll be busy with another project, then I am flying to Pittsburgh for a week.
What happens to the RO(G)-grading idea with the move to global equivariance?
Along the general lines laid out at orbifold cohomology and specifically the setup on slide 78 here I am looking at orbifolds $\mathcal{X}$ equipped with a faithful morphism to the delooping groupoid $\mathbf{B} Pin(5)^\flat$ of the discrete group underlying $Pin(5)$. Then equivariant cocycles are maps in the slice
$\array{ \mathcal{X} && \longrightarrow&& S^4\sslash Pin(5)^\flat \\ & \searrow && \swarrow \\ && \mathbf{B} Pin(5)^\flat }$Now $Pin(5)^{\flat}$ is of course not finite. But by $\mathcal{X}$ being an orbifold, we have that around any of its singularities, any given morphism on the left will factor through the inclusion (under $\mathbf{B}$) of a finite subgroup $G$ of $Pin(5)$. By pulling back along that inclusion in the above triangle diagram, we see then that in the vicinity of that singularity the “global” cocycle reduces to one in $G$-equivariant cohomology.
This took me a while to understand: That an orbifold regarded in the faithful slice in this fashion has attached to each of its singularities the further information of which “kind of charge” may be found inside this singularity, namely the choice of how the isotropy group of the singularity is to act on the coefficient 4-sphere.
This is an effect of an aspect of “global” homotopy theory.
Thanks, Urs. I will be free to focus more on this stuff in a week and a bit.
Having been mired down in other things for a while, it’s great to revisit this story with all of its parts.
You sent me to the one typo I saw
Cohomotpy (slide 78)
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