@JonBeardsley I believe that's correct, yes.

Hey guys... dumb question -- if X is a connective spectrum, then the MU-ANSS for X converges to the homotopy of X, right? Even if X is not finite?

I feel like this follows from one of Bousfield's papers.

I just can't remember.

Hi! For homology, given a space X and a cover of X with two open subsets U_1 and U_2 one has the Mayer-Vietoris sequence. Is there a similar sequence for an arbitrary covering, and if so, is there any reference?

I fixed it somewhat

maybe a mayer-vietoris spectral sequence is needed for a covering of X by more than 2 open subsets?

@Dedalus yes, precisely

Bott and Tu basically say that the Mayer–Vietoris theorem is a special case of a spectral sequence

Can the Hopf map $\nu$ be at all related to the attaching map for the top cell of $\mathbb{CP}^3$ (rather than having to do with quaternions)?

this is this great paper of mosher called some stable homotopy of complex projective space where he tells you about the attaching maps in the even stable cell decomposition of CP^infty out through quite a range

in particular, there's proposition 5.2 in there

re: convergence, i think you can use connectivity type arguments to show that the MU-ASS converges for bounded-below spectra with finitely many cells in any given degree

Sorry @EricPeterson can you explain proposition 5.2 to me?

or rather, why that's of interest?

what he's describing in that section is the atiyah-hirzebruch spectral sequence for H^*(CP^infty; pi_* S) ==> pi_* CP^infty, though he's indexed it pretty funnily. differentials in that spectral sequence record attaching maps, and he's giving a formula for which (n+2)-cells are attached to the n-cell beneath them along a nu

the proposition is a little cryptic, but maybe with that explanation you can tell what it's trying to say. here it is: cl.ly/image/1E140q1i141y

okay. so for starters at least... it seems plausible that the attaching map for the top cell of CP^3 is related to \nu

the reason i bring it up, perhaps obviously, is that i'm wondering why akhil's DA(1) kills the second obstruction to complex orientation

when DA(1) seems to be more or less built from \eta and \nu, whereas the second obstruction to complex orientation should be the attaching map of the top cell of CP^3

i'll to have to look harder at this paper by mosher tho

thanks for the ref.

:) gr8 reason to think about this paper

oh i see now. okay, he's taking the cellular filtration on CP^\infty and just doing it's associated sseq after applying pi_*^s

yes

(which is indeed the AHSS thatyou describe lol)

my cat is really trying to get involved in this conversation. but i don't think she knows anything about stable homotopy groups.

@EricPeterson sorry, do you know specifically what $\beta_n$ is for him?

he doesn't seem to define it clearly.

maybe it's in this paper of Toda he references though.

it's the homology class representing the 2n-cell

oh! ok

(you should check that to make sure i'm not off by 1, there should be a differential d1 b2 = eta b1)

that's right

@EricPeterson So, the spectral sequence is constructed from some double complex? I am trying to think if I can get the spectral sequence from an exact couple somehow, but I can't see it.

@Dedalus imho every spectral sequence is constructed from a filtration. this one comes from constructing the simplicial space (or cech object) associated to the cover and taking the skeletal filtration. so, E^1_{s, t} in the exact couple comes from the degree t homology of the (nontrivial) s-fold intersections of the spaces in the cover (and D^1_{s, t} comes from the degree t homology of the realization of the s-skeleton)

but the filtration is the natural object, not the exact couple

Hi everybody, finally I can speak :)

Thank you for all your help!

@SanathDevalapurkar That's a very broad question. If you are asking whether there are any areas of number theory where the language and concepts of category theory are employed, then there are many (most of the areas involving modern algebraic geometry or homological algebra, for example). If you are asking whether there are theorems of pure category theory with immediate consequences in number theory, I'm not aware of any.

That seems more likely.

Although it's sort of unclear to me how one can precisely separate these two notions. The only real number theory I know is algebraic geometry. Just, weird algebraic geometry.

Well, there is a whole branch called "analytic number theory".

Haha, I know. =P

Dang ol' integrals.

@SanathDevalapurkar I guess, again, that this depends a little on your perspective. The structure of the ring of integers (such as being Noetherian, having homological dimension 1, etc etc) has a number of consequences (the decomposition of fields according to characteristic, the classification of finite extensions of them, simple expressions in universal coefficient sequences over the integers, etc etc).

The Mordell conjecture, and Faltings' theorem, give hard bounds on the number of points that certain types of algebraic variety can have.

But without a more specific target, it's sometimes hard to give something that feels like a satisfactory answer to "Is field X important to field Y?"

Hey @TylerLawson can we think of Thom spectra as being twisted tensor products in some sense? In the way that for a fibration $F\to E\to B$ we can find an equivalence between the chains on $E$ and a sort of (potentially twisted) tensor product of chains on $F$ and $B$?

how far is this tyler answer mathoverflow.net/a/138228/1094 from being an answer to that

oh lordy, this thing

maybe i'll actually be able to understand this question now, since i spent some time hitting myself in the head with abghr

Yeah, actually I think your question is intimately connected with exactly what I've been going on and on about the past few days.

Well, I guess that's not a surprise, since IIRC, it's basically a question I asked you to start with, right?

It's only taken me about a year to catch up.

(sort of)

i don't remember, but yeah, it's something you and i were both interested in at some point

Yeah, I think we were trying to figure out these X(n) spectra a little better. Or at least, that's where I entered the game.

Hm, yeah, it hadn't quite occurred to me that this HZ-orientability hypothesis would play the role of giving an isomorphism on the E_2 page of the AHSS

That's a nice thing to notice.

Are you familiar with this paper of E.H. Brown where he does this for singular chains and fibrations?

maths.ed.ac.uk/~aar/papers/brown6.pdf

I mean, that's precisely a "twisting cocycle"

no, i'm not

1959, gosh

And so if we think about that in terms of a sphere spectrum bundle over a space $\mathbb{S}\to E\to B$, I dunno.

do you happen to know the answer for your question if you're dealing with a finite dimensional complex vector bundle over a space? that just comes down to knowing the cell structure of the thom space.

no, i don't

Yeah, I don't either. I mean, even supposing we've got an $S^n$-bundle over a space. Hm...

i'm pretty confident that knowing space-level cell structures is strictly harder than knowing spectrum-level cell structures in every situation

Ah fair enough. Even so, at least restricting oneself just to Thom spectra coming from complex vector bundles...might (?)... make things easier?

I dunno.

i was hoping to leverage that, in the form of james periodicity. mosher succeeds in implementing basically every idea i had, actually, and none of them allow him to perform anything like an inductive argument (if only he had sufficient information about pi_* S)

Haha.

Sufficient meaning all the information?

It would seem that with $\pi_\ast S$, it's either all or nothing.

yes, even granting that

@AaronMazel-Gee i'm bringing tomatoes to your YTM talk. just in case.

just kidding.

I don't want to be a cyber-bully.

fyi his talk looks to be great

;-)

I know

Oh, I don't think I realized that Tyler finished this answer.

against your comment on the answer?

Oh.

Fascinating.

:)

Hrmmmm. Okay. Yeah. I don't entirely understand it, but it smells sensible.

But... yeah, so, one thing to say is: they have the same filtration quotients for their cellular filtrations? Is that truE?

Sorry, by they I mean the suspension spectrum and the Thom spectrum.

that's true. it's even true without the orientability hypothesis, which i thought might be necessary but it isn't

Which of course makes sense, considering that if $\phi$ is the zero map, you just get the suspension spectrum of $X$ back.

Right okay, of course. Since $P$ looks like $G$ locally.... hmmmm very interesting.