@skd i specifically think no. the nilpotence order of eta in bo comes from differentials in the fixed point spectral sequence; as far as the algebra is concerned, eta looks to be non-nilpotent

well idk maybe that's speaking strongly, but i don't know how to convincingly phrase those differentials as "ah, this is just naturality and some other nearby algebra"

@EricPeterson that makes sense, thanks

the tmf book says that DA(1) comes from some 8-fold cover of M_{tmf}, which is what motivated that question

What are some simple illustrative examples/counterexamples of $\infty$-toposes for all the businesses of Hypercompleteness and convergence of Whitehead/Postnikov towers?

etc...

I have some trouble with this part in HTT and thought it might be easier to go through with examples...

I just realized that in the document data for HTT (croppedtopoi.pdf), the author is listed as "The Boss"...

@SaalHardali there's a comment on nForum by Marc Hoyois pointing to Example 2.1.30 of math.uiuc.edu/K-theory/305/nowmovo.pdf as an example of a hypercomplete oo-topos where Postnikov towers don't converge for all objects.

Idiot question. If I have a square satisfying Beck-Chevalley for one corner (g^* f_! \cong k_! j^*), is it true that Beck-Chevalley holds for the other corners? It's probably true, but I'm getting something wrong in my computation.

Hi all, I've a quick question: using the isomorphism \pi_*(MSpin)=\pi_*(ko) for *<=7 (am I right? I can't find it anywhere but everyone seems to use it) , If I want to compute low dimensional spin bordism groups MSpin_*(X) for a space X I might just compute ko_*(X). To this end, what is the coefficient ring for the cohomology ko^*(pt.) ? (because I'd like to make use of the multiplictive structure of the cohomology + the pairing of AHSS)

@LuigiM ko_* = Z[eta, alpha, beta]/(2*eta, eta^3, eta*alpha, alpha^2 - 4*beta) where eta, alpha, beta are in degree 1, 4, 8 respectively

That's a reasonably easy calculation (at least 2-locally, and for the additive structure) once you know you can use change-of-rings to get the E_2 page of the Adams spectral sequence for $\pi_* ko$ being Ext_{A(1)} (F_2, F_2).

You can just write down a resolution

I see, thanks to both of you! I'll meditate on these comments a little bit

rognes has a document on this, which is pretty nice: uio.no/studier/emner/matnat/math/MAT9580/v15/…

I should probably comment that drawing a resolution is probably a more accurate and helpful phrasing

one can also compute pi_* KO via the complex conjugation hfpss, and then get pi_* ko from that

@skd nice notes!

@LuigiM the same computation (of pi_* ko) is done somewhere in ch. 3 of the green book

i'd strongly recommend trying it for yourself first

Ok I start to see your points, this lead to another question. in order to make use of multiplicative structure of the AHSS and the pairing should I compute KO instead of ko? because ko_=ko^{-} so if I need to pair (for example) element in the first quadrant of the homological AHSS; I'd have an always trivial pairing, since ko^* (for positive *) is zero

@PeterNelson I agree with that, but before jumping to the computations I'd like to lay out a strategy and understand if these machineries can help me or not

perhaps i'm being silly, but the E-(co)homology of something is generally very different from the t_0 E-(co)homology of that thing (here t_0 E is the connective cover of E)

exactly, but I do believe that the positive degree homology should be the same right?

for instance, if E is the Anderson dualizing spectrum I_Z, then t_0 E = HZ, but the Z-(co)homology of anything is pretty different from the homotopy of the Anderson dual

my strategy is to make use of this and exploit the fact that KO has non trivial positive cohomology to have possible interesting pairings in the relevant AHSSeqs

since I'm not an expert in these tools, my intention was to ask whether this approach makes sense or I'm overlooking something