@YanZou I tried looking on KBrown "Cohomology of Groups" for the computations you mention but with no success. I used google books OCR to search for "Dihedral" but the results shown weren't about its (co)homology groups.

Given a spectrum map $\alpha\in Map(S^k,E)$ for an $A_\infty$-ring spectrum $E$, is there a concrete description of its adjoint in $Map_{A_\infty}(F(S^k),E)$? Like, in terms of homotopy classes?

@Riccardo I can't remember, but it might be in the exercises.

@JonBeardsley I'm not sure I understand your question. What do you mean by a "concrete description"?

@DenisNardin perhaps I am mistaken, but the free A-infty algebra on S^k has a nice description right?

I.e. it splits as a wedge of spheres (at least additively?)

Sure, its underlying spectrum is $\bigvee_i S^{ki}$

My first guess was gonna be "it sends the generator in degree nk \alpha^k" but I guess that doesn't work if k is odd... do I need a Massey product?

So can I write the map down as some kind of sequence of homotopy classes of E?

@Dylan Why doesn't that work?

Here's my example: do this with eta mapping to S, you get X(2), but eta is nilpotent.

Sorry, attach an associative cell along this asjojnt

Ack! On my phone.

But it could be that I'm just failing to understand the process of attaching an A_infty cell

If you are satisfied with a description of the underlying map of spectra, yes

Oh hold on you want to take the pushout in A_∞-rings? Then you need all the coherence data

(At least a priori, I'd love if someone took out of a hat a way to ignore them)

you could convert it back to the old system where it's associative on the point-set level

I guess I don't have an immediate counterexample... but if you smash Free(S^1) with like \hf_p you should get the free A_\infty-\hf_p-algebra on a generator in degree 1. But the square on the generator in degree 1 (in homology) is zero (if p>2) by graded commutativity? or is it not graded commutative because I took the A_infty thing?

yeah, whoops... hmmm

@TylerLawson But then, don't you need to take some kind of cofibrant replacement in order to take a pushout that does the same thing?

yep

but the tradeoff is that the pushout process itself tends to be easier to analyze

and if you're doing things like attaching cells then you're typically starting with some kind of cofibration F(S^k) -> F(D^{k+1}) anyway

Hrmmmm.

It's weird, because in Mike Hopkins' thesis he has this neat cellular description of X(2) (at least, localized at 2), and I'm trying to relate this to the idea of attaching an A_\infty-cell.

Or like, Ravenel describes T(1) (which is basically to X(2), at least for p=2, what BP is to MU), as being basically S u_{\alpha_1} e^2 u_{\alpha 1} e^4 etc. etc. where one is continually adding cells of even dimension along \alpha_1