@SanathDevalapurkar it's related by a sequence of also-nontrivial facts. L_n-localization acts by rationalization on eilenberg-mac lane spectra, so a (spectral) map from Z/p into gl_1 E_n is sent to zero by postcomposition with gl_1 E_n --> L_n gl_1 E_n, so EinftySpaces(Z/p, GL_1 E_n) = Spectra(HZ/p, gl_1 En) = Spectra(HZ/p, F). then pi_j Spectra(HZ/p, F) = pi_0 Spectra(S^j HZ/p, F), and since F is n-cotruncated (by AHR 4.11), that last thing is = pi_0 Spectra((S^j HZ/p)(-infty, n], F)

if j exceeds n, then the source becomes contractible and the mapping space collapses

lurie's conjecture in the MO answer is an assertion about the Einfty space associated to Spectra(HZ/p, F), which i guess may have lots more coconnective information as a spectrum? but as a space he's guessing that it's a K(Z/p, n)

i actually wonder if that's a theorem now, last time i was in boston i heard that there had been a successful identification of the 'discrepancy spectrum' (which i think is F) (or maybe its connective truncation) with (a shift of) the brown-comenetz dualizing spectrum (or maybe its connective truncation)

@EricPeterson Peter's wife is named Maya (maybe ma1ja?)

a mild stretch

You're generous

does the following construction have a name? start with a connected space X. write down the natural map X -> B H_1(X, Z) inducing an isomorphism on H_1. take its homotopy fiber X_1. write down the natural map X_1 -> B H_2(X_1, Z) inducing an isomorphism on H_2. take its homotopy fiber. etc.

in other words, it's like the whitehead tower, but you're trying to kill the homology of X rather than its homotopy

in particular, does the sequence of homology groups H_{n+1}(X_n, Z) you get this way have a name? (as far as I can tell it is not something familiar, or at least not to me)

in particular in particular, is it known what happens when you apply this construction to X = BS_n, at least stably in n?

(i have a vague suspicion that the answer, stably in n, is that you get the stable homotopy groups of spheres, and i'm trying to figure out whether this follows from barratt-priddy-quillen-segal or is a different statement)

S^j is suspension, and X(-infty, n] is my favorite notation for the object under X which is n-coconnective and whose homotopy groups agree with those of X up to & including n

(or: the fiber of the map X --> X(-infty, n] is n-connected & could be reasonably called X(n, infty))

yeah, which is also F

yes, & i think that whole discussion is worth reading

unfortunately my cloud paste is gone. i was just asking for someone (charles, in particular) to explain the structure of gl_1 KO to me

the email thread, restored: cl.ly/2d1J2C1G1I3j . unfortunately i was asking a question that was premised on something false, and charles's reply is mostly about explaining the correct premise. still worth reading

i'm still tickled that eta in pi_* S acting on "eta" in pi_* gl_1 S has eta * "eta" = 0. bob bruner sent me? tried to send me? a paper by waldhausen that roughly said that the (pi_* S)-module structure of pi_* gl_1 R is essentially the same as that of pi_* R, except for etas, which appear and disappear only as they need to to make homotopy theory self-consistent

i have not yet tried to understand that claim at all

haha wait wut

i have been spending my energy on making a Web App recently, chat this afternoon is reminding me that math can also be fun

heh. allegedly.

what do you mean by "etas"?

as in, plural etas.

i mean the collective action of the map pi_* gl_1 R (x) pi_1 S --> pi_{*+1} gl_1 R as * varies

rather, that's what i thought was meant. i really haven't read the waldhausen paper so i can't actually honestly say anything

but are there multiple elements i might refer to as "eta"?

it's not really relevant to your actual statement

the plural was only intended to reference the many elements against which you can multiply eta. eta itself is singular

o ok

@QiaochuYuan I don't know if that construction has a name, sorry. I think that in the case of S_n (stably) you can, for n >> 0, inductively show that the map BS_n -> (BS_n)^+ induces homology isomorphisms at all stages of this tower, and conclude that you're effectively calculating the Whitehead tower of (BS_n)^+

so yes, for large n you're recovering the (positive-degree) stable homotopy groups of spheres due to homological stability for symmetric groups + Barratt-Priddy-Quillen

@EricPeterson do you happen to remember which Waldhausen paper that was?

also, the following result about etas and GL_1(R) is highly amusing: arxiv.org/pdf/math/0405079v1.pdf

@EricPeterson I would be VERY surprised if enough was known about the power operations on E_n to employ Charles' approach, but maybe others have been more clever.

(for the record/others, Charles' approach to h=1,2 is here: math.uiuc.edu/~rezk/power-ops-ht-2.pdf)

@TylerLawson from bob's email: §2 of "The map BSG -> A(X) -> QS^0", Bökstedt and Waldhausen. again, I haven't looked for myself

@PeterNelson tobi tried to tell me how the Bostonians pulled it off, but I forget what he said. Maybe he'll send you a Private Communication? In any case it wasn't by Charles's method

maybe said private communication could be cc'ed if it happens

@TylerLawson :) sure

@EricPeterson thanks!

Also re: the discrepancy spectrum result, mike wanted to know (last summer) whether the cech spectral sequence associated to the chromatic fracture hypercube gave a genuinely new way of conputing stable stable stems or if we would recognize it from somewhere else. That's soecifically whar Tobi and I talked about, his suspicion (last summer) was that we'd recognize it from somewhere else

@Tyler: great, thanks! is this a special property of BS_n or is it generally true that this construction ends up recovering the whitehead tower of the plus construction?

wow... Hartshorne, May, and Thomason all had the same advisor (Moore)

huh, so I guess that would make Hartshorne my supervisor's third cousin once removed

@QiaochuYuan what was necessary, so far as I can tell, is that it's the commutator subgroup [S_n, S_n] which is perfect. I had to use this to conclude that the fiber of X^+ -> K(H_1,1) was an (X_1)^+

you know what, that's not correct. to take a homology isomorphism X -> Y and conclude that fib(X -> K(A,1)) -> fib(Y -> K(A,1)) is a homology isomorphism, you need to know more; the assumptions only allow you to conclude something about the maps H_*(A; H_*(fib_1)) -> H_*(A; H_*(fib_2))

all right. I think that by being more explicit with a cell description, this does work: if X has perfect commutator subgroup of pi_1(X), then you can arange a map X -> X^+ so that taking the fiber of the map out to K(H_1 X, 1) gives a map Y -> Y^+.

sorry for the flailing around

@TylerLawson that schlictkrull paper is really really fascinating....