@ShayBenMoshe The way to get intuition for the proof is to look at Ravenel's proof for odd primes

$\Omega_{\mathbb{O}}$ is not really an analogue of $E_4$ as much as an auxiliary spectrum that should be used in the construction of that analogue, but they realized they did not need to go further with the construction (at least according to Hopkins)

Thanks, I will take a look into his paper on the odd case, although I am not sure that this will give me the kind of intuition I hope to have

Do you mean that $\Omega_{\mathbb O}$ was originally perceived to be part of a construction of an analogue of $E_4$, but they stopped in the middle because they figured out it suffices for what they need?

Yeah, at least that's what Hopkins said in a talk I attended

I see

The thing is that I don't understand why even look at this kind ($E_4$ or $\Omega_{\mathbb O}$) of spectra for this purpose in the first place

I will take a look at Ravenel's paper on the odd case, if you're saying that might help

But if anyone has another, hopefully conceptual, explanation which hints why this is a good idea, I'd love to hear that

Basically you want to say that some classes in the ANSS are not permanent cycles. To do so you compare the ANSS with the K(n)-local E_n-based spectral sequence, which is equivalent to the HFPSS for the action of the Morita stabilizer group on E_n

Essentially you're using that if the image of a class is not a permanent cycle under a map of SS, the original class cannot be a permanent cycle either

Yes, I understand the idea of the proof

But I lack the motivation for the specific spectrum they use for the comparison of SS

If you think geometrically, this is comparing the descent SS for the stack of formal groups to the descent spectral sequence of the completion of that stack at a certain point

It would have been a certain point if it was $E_4$, but how is $\Omega_{\mathbb O}$ a certain point of the stack?

Well, $E_4$ is the original motivation, apparently that was what they were building to, but it turned out to be unnecessary

Also why is it that the $HF_2$-ASS the descent SS for the whole stack of fg? Is it because the neighborhood of the point at height infinity sees all other points?

No, that's the ANSS, the $MU$-based SS. The first step of the proof is reducing from the $H\mathbb{F}_2$-based SS to the $MU$-based one

Well, but why $E_4$ then? Why not say $E_3$?

Cool, I'll look into that step

Oh is that the part of the algebraic detection? where they do this span of SS?

If I recall correctly, Hopkins said that they tried various $E_n$s and 4 was just the smallest for which they managed to run the proof

I see

It is related to $C_8$ being a subgroup of the Morava Stabilizer group

If you look at the proof the $C_4$-cohomology is just not sparse enough to run the proof

I see

And do you have an intuition for why $\Omega_{\mathbb O}$ is (a part of) an analogue to $E_4$? I mean, how would you come up with that alternative?

(I realize that being the Hill, Hopkins and Ravenel helps with coming up with good ideas like the norm and so on, but there might be a reasonable explanation a posteriori)

i think this is true: if you want to detect eta^2, it suffices to look at KO = E_1^{hC_2}. but eta^2 is already detected in MU_R^{hC_2}, and there is a C_2-equiv map MU_R -> E_1 = K_R. so since the calculation of E_4^{hC_8} is pretty hard, you can instead look at the C_2-equiv map MU_R -> E_4, which norms up to a C_8-equiv map MU^(C_8) -> E_4. then the desired kervaire elements are already detected in the C_8-htpy fixed points of MU^(C_8), which is easier to calculate

aha, that's a nice explanation

just to be sure, the map MU_R -> K_R, is like the C_2-upgraded complex orientation right? (combining the MU->KU and MO->KO maps equivariantly)

@ShayBenMoshe I don't want to put words in skd's mouth, but I'm pretty sure that's the case (also, that was a very nice explanation)

Cool. Thanks to both of you, that conversation was really useful :)

(also if anyone has even more intuition, I'd love to hear that too)

There is no map MO—->KO...

presumably @ShayBenMoshe meant to write MU^{hC_2} instead of MO

@skd This does work, but somehow I also thought that the geometric fixed points of K_R is KO, isn't it? (if it is, then taking geometric fixed points to MU_R->K_R should give MO->KO)

The geometric fixed points of KU are 0, the fixed points and homotopy fixed points are both KO

Ok, good to know that..

Does anyone have a good reference for these kind of statements and constructions? I'm struggling to find something which doesn't assume that I know all of it

HHR itself, the sketch of the proof by Hill, and the notes from the Talbot workshop on the proof are all nice references

@DennisNardin I like your "Bredon $G$-spectra" terminology suggestion, I may use it in the future.

@DenisNardin Using equivariant Atiyah duality, can one get something like ”equivariant Verdier duality”?

Does anyone know if it's possible to give the J-homomorphism $BO\to Pic(S)$ as the K-theory of a map of (bi?)permutative categories?

In the category of spectra when is it true that $Hom(X,\mathbb{S}) \otimes Z \cong Hom(X, Z)$? Its clearly true when $X$ is dualizable. I'm more interested in optimal conditions on $Z$ which make this true for all $X$.

In modules over a ring i think this is true when $Z$ is flat for example.

(i don't know if this is an optimal condition though).

@Dedalus I'd think that the implication would go in the other direction. After all classical Verdier duality (for local systems of spectra) is stronger than Atiyah duality