@JonathanBeardsley I think I might be misunderstanding something, but it sounds like the map $\ast \to GL_1(S)$ you described isn't $O$-equivariant so I don't see why it would induce a map on orbits.

@SaulGlasman Well. The usual definition as for instance in ncatlab.org/nlab/show/sketch , i.e., a category with prescribed cones and cocones

@PaulVanKoughnett that might still be a good option.

@DenisNardin $\pi _n (S^m) = \Bbb Z $ if $m \neq 2$ and I presently am working on $\pi _1 (S^2 \setminus M ,E)$ that $E$ is a specified point, and $M$ is a real and unknown subset of $S^2$ and I do not know what is the $\pi _1 (S^2 \setminus M ,E)$ ? and I can not understand "wedge of S^{n-1}'s and then π_n is a sum of Z/2's" because I am new in Homotopy!

@AlirezaBadali I am afraid that $\pi_n(S^m)$ is equal to $\mathbb{Z}$ only when $n=m$. I strongly suggest that you familiarize yourself with at least the first chapter of Hatcher's book, for the $\pi_1$ and the fourth chapter for the higher $\pi_n$'s.

@DenisNardin I thank you Denis.

@DenisNardin That is true stabley. $\pi_3(S^2)=\mathbb{Z}$ as well. It is only stabley that the first hopf map is two torsion. Also, $\pi_{4n-1}(S^{2n})$ has a nontrivial rationalization.

@Riccardo It is possible people didn't write them down. If you want to check your work maybe write to Bob Bruner? Or you could peruse his website.

But why doesn't work MathJax here?

@SeanTilson Yeah, I forgot to mention that $\pi_{4n-1}(S^{2n})\otimes \mathbb{Q}=\mathbb{Q}$, although I think that it almost never happens that there is no torsion

@AlirezaBadali If you look at the description of the room you will find the instructions to make it work

But is $\pi _n (S^2)$ a Group? particularly if $n=1$

@AlirezaBadali $\pi_n(X)$ is always a group for $n\ge 1$ and an abelian group for $n\ge 2$ for every space $X$.

@DenisNardin I thought being group $\pi _1(S^2) $ is a open problem.

No $\pi_1(S^2)$ is well known to be the trivial group (it follows from the van Kampen theorem, or the cellular approximation theorem). Where do you have this information? Again, you should probably at the very list skim Hatcher's book.

In particular the group operation on $\pi_1$ is concatenation of loops

@DenisNardin Because I was so far from Mathematics for $14$ years and I had heard it in bachelor course. But what time is it now in the USA? anyway I thank you so much Denis. and goodbye I have to go!

@SeanTilson I'll try to write down my computation and maybe ask a question here before mailing directly to him, because I'm not an expert when it comes to ass and I fear I'll do stupid errors:)

@user40276 Ah, I didn't know that terminology. I don't know if this is the answer you're looking for, but once you have the theory of limits and colimits in quasicategories set up, you can write down the analogous definition without any additional difficulty

Ummm Hi

are there notes from this year's european talbot?

I thought I'd heard in this room before that we don't really know how to do formal spectral schemes. But Chapter 8 of SAG is Formal Spectral Algebraic Geometry. Can someone explain to me what the subtleties here are? (why Jacob's defn might not always give you what you want, etc.)

one point to make is that in formal geometry you consider things topologized by powers of an ideal

but when you quotient an E_oo-ring R by an ideal (generated by a regular sequence), the result usually won't be another E_oo-ring

for instance, if you're to look at, say, the Lubin-Tate space Spf W(F_p^n)[[u_1, ..., u_{n-1}]], with maximal ideal m = (p, u_1, ..., u_{n-1}), this isn't the correct thing to do since things mod p are rarely E_oo-rings

in that chapter of sag, lurie essentially defines Spf R for an adic E_oo-ring R (which is an E_oo-ring along with a finitely generated ideal m of definition) to be Spf pi_0 R along with a sheaf of E_oo-rings given by composing O_{Spec R} with the m-completion functor

and so the key problem is defining m-completion, which is done in section 7.3

@OmarAntolín-Camarena hehe you're totally right :(

@YuriSulyma everyone has heard me complain about this before, so I'll keep it brief, but you may know that the moduli stack of formal groups has a height filtration that shows up in the stable homotopy groups of spheres (and in the category of finite complexes itself)

and this is sort of the starting point of chromatic homotopy theory

so one might expect that MU would naturally show up as some kind of ring spectrum controlling "spectral formal geometry," thereby giving us a "reason" for formal groups showing up in the stable homotopy groups

as far as I can tell, no existing definition of "derived formal geometry" really sees this structure

that is NOT TO SAY that existing constructions, especially Lurie's, are not useful. they're very useful, as Lurie has demonstrated. but this has always been a sort of bummer, in my book.

I think the fact that the K(n) spectra, and BP, are not E_\infty, suggests that working in the E_\infty world might actually be TOO strong (to get the chromatic structure naturaly). But, again, I imagine a lot of people would disagree with me.

What are some good references for computing ordinary cohomology of spectra?

I would like to compute H^*(X smash Y; F_2) as a module over the Steenrod algebra, where X and Y are spectra whose Steenrod actions I know

but I don't know whether the Künneth theorem holds in this context

I tried Googling for references, but can't find anything