i have a question that i haven't really thought through yet, so apologies in advance: can the cell structure of the thom spectrum of a spherical bundle X --> BGL_1 S be determined from sufficient, oracular knowledge of that of X and the behavior of the classifying map? what exactly would it take?
possibly a sane phrasing of this is: given complete knowledge of the atiyah-hirzebruch spectral sequence H_*(X; pi_* S) ==> pi_* X and some unspecified amount of knowledge of the map X --> BGL_1 S, can the differentials be determined in H_*(T(X); pi_* S) ==> pi_* T(X)
sure, maybe suppose that the spherical bundle is HZ-oriented to start
the most basic question to ask is what the thom spectrum of a map S^n --> BGL_1 S is. is it an M_0 moore spectrum on the classified homotopy element of BGL_1 S = Loops^infty S^infty [1, infty) ? (is that equality even right?)
I mean in the 'old days' I think you built the Thom spectrum by filtering BGL_1S (usually you had some explicit model of this in mind and the filtration is analogous to the one for BO(n)) and then getting a corresponding filtration of the map f: X ---> BGL_1S. Then pull back the spherical fibrations you get, take Thom complexes, and all these together give the spectrum
but that description doesn't seem immediately ready-made for the types of applications it seems like you're thinking about
that is how you do it in the old days; you simultaneously use that you can filter the source by finite dimensional skeleta, and those compact objects factor through finite stages of the BGL_1 S filtration
here's another way of thinking about the question, which is maybe the same way
the bigger question, at least. not your specific one.
we're asking about extending n-buds on $E_\ast$ right? well, we can always do this algebraically, so what's the precise difference between algebraic extensions of n-buds and topological extensions of n-orientations?
the difference between the algebra and the algebraic topology there is the difference between H_* CP^infty and the spectral sequence H_* CP^infty (x)_* E_* ==> E_* CP^infty, which is (in a totally inaccessible way) controlled by the answer to my question. they're related 4 sure
well steenrod algebra structure would be something, but i worry it might be in an orthogonal direction. steenrod algebra structure + all n-order structure for n > 1 is supposed to determine 2-adic homotopy type. i can phrase this question in that language instead of the pi_* S one, but i think pi_* S is what i want, since fib(bgl_1 S --> bgl_1 HZ) has something to do with pi_* S
that's exactly what i want and what i mean, the steenrod structure would let me do it, provided a lot of extra fuss with extra things past the E_2 page in the HF_p adams ss. maybe that will be useful when it comes time to compute something, but for the moment just framing the answer shouldn't necessitate steenrod operations, i wouldn't think, provided i'm willing to tolerate having pi_* S lying around as potential input
a formula for the attaching map wedge_alpha S^n --> T^(n) --> T^(n+1) for a thom spectrum T = Th(X --f-> BGL_1) in terms of a formula for the attaching map wedge_alpha S^n --> X^(n) --> X^(n+1) and some kind of description of the map f
i recently bought the green book, because i thought i was going to get involved in a project that used computations in it seriously, so i'd better own a copy
now i'm thinking i might not get involved after all, but i should still take the time to read a sizable chunk of it
where $H^\ast(\Pi,n;G)$ is the cohomology of the Eilenberg MacLane space
$K(\Pi, n)$
moreover, they say that if we consider the product on $B\Pi\equiv K(\Pi,1)$ given by shuffles, $\ast:B\Pi\times B\Pi\to B\Pi$, then $H^3(B\Pi,im(\ast); G)$ is always trivial
and this last theorem is driving me nuts
their proof is pretty confusing
and i'm super stuck, and kind of desperate. because i want to write down the same theorem for group schemes
o, the first two are easy if you believe in delooping; an extension G --> E --> Pi deloops to give Pi --> BG and then to BPi --> BBG, and given a map BPi --> BBG you can use the loop sequence to unwind to an exact sequence G --> E --> Pi
similarly for the abelian extensions, except there you're assuming you can deloop twice, which is not an accident
@ArnavTripathy best to highlight @AaronMazel-Gee
why should you have to hit [tab] before hitting [enter] to complete a name
so funny story, you need X and Y to be E-local for some homology theory E in order to use goerss--hopkins. and you need a handle on the homology cooperations. so you could try to use stable homotopy, but then your homology cooperations are sort of intractable
@ArnavTripathy arnav, i said what their obstruction groups were above for your question. there may be some hypotheses that aaron points out. I can tell you the appropriate references.
@JonBeardsley that's the only definition of group cohomology i know: H^n_{gp}(G; M) = H^n_{spaces}(BG; Z; M), where M is taken to be a twisted coefficient system. if M is a trivial system for G, then this is H^n_{spaces}(BG; M), and H^2 there is classified by maps [BG, K(M, 2)]
hmm, does anyone know how a functor of DG categories being a Morita equivalence categories compares to just inducing an equivalence of the homotopy categories
@EricPeterson so this appears to follow from a certain isomorphism of groups (probably of rings....) $H^4(K(\Pi,2);G)\cong Hom(\Gamma(\Pi),G)$ where $\Gamma(-)$ is Whitehead's quadratic functor, that takes a group to something which behaves like quadratic functions on that group
it's sort of fascinating. I'm trying to determine what the right analog of this functor should be for group schemes, or at least, what thing I can plug in that will make the above isomorphism hold for group schemes (er.... stacks?). I think that in the case of $\mathbb{G}_a$ it might be something like degree 2 polynomials (or the sub-stack whose global sections is such)
