skd

where can i find an explicit description of the map of operads Lie[n-1] -> E_n (namely, the map at each arity)? by this i mean the map of spectral operads which is the shift + koszul dual of the map E_n -> E_oo (so Lie is the spectral lie operad)

@TimCampion in char 0, i find it useful to think about E_n-algebras (n>1) in chain complexes in terms of their power operations, i.e., as a commutative algebra object + a poisson bracket of degree (n-1). as n -> infty, this poisson bracket increases in degree until it ceases to be relevant. (this is related to why the E_n-operad is, up to an important shift, koszul self-dual for finite n; but its koszul dual for n=infty is the lie operad)

@DustinClausen robert says (ku here is the adams summand): knowing the homology of ku as your input: then S/(p.v1) --> Fp and ku --> Fp can be tensored together using the ring structure on Fp. By the previous construction of S/p,v1 this will be an iso on Fp homology. Turning it around you get that v_1 on S/p induces v_1 on ku/p. The homology of ku can be computed starting with its homotopy groups by modding out by p,v1 and then observing you get an Fp and working backwards

@DenisNardin robert burklund wanted me to say the following (for p odd): the 2p-skeleton of HF_p has 6 cells. the top 4 cells are two copies of the S/p (this needs \pi_1 = 0) and the bottom two are S/p, so you get a pair of endomorphisms of S/p in dimensions that differ by 1, one of which is \alpha_1 and other is v1

@ManuelRivera i think the terminology is analogous to the situation in complex geometry, where a variety is called calabi-yau if it has a nonvanishing holomorphic n-form/trivial canonical bundle (but such a differential form/trivialization isn't generally part of the data)

oops yes, sorry. that doesn't make sense (i was just thinking about concatenating linearly ordered sets)

re kiran's question, here's a related thing. the category of parasimplices has a nonunital (nonsymmetric) monoidal structure: if S and S' are parasimplices, define S (+) S' to be the parasimplex with underlying set S u S' (disj union), and if x, y in S u S', say x leq y if either x, y in S and x leq y, or x, y in S' and x leq y, or x in S, y in S'. the Z-action comes from that on S and S'.

@AaronMazel-Gee fwiw another way of understanding factorization homology over S^1/thh as a colimit over the paracyclic category is as follows: the exit path category of Ran(S^1) is equivalent to the opposite of the category of parasimplices with surjections as morphisms. using this identification, one can translate what it means to be an E_{S^1}-algebra (i.e., the factorizability condition). then the covers of S^1 give you the cyclotomic structure on thh (see arxiv.org/abs/1710.06409)

yeah, you're right. i was just stating it for E_n-modules because i didn't want to overstep and claim something wrong

this is supposed to be a pd-analogue of a result of hopkins-lurie at height 1, which says that if you take the free E_oo-KU_p-algebra on one generator x, and kill theta(x) in an E_oo-way, then you get the flat affine line KU_p[x] (see the first lecture in math.uchicago.edu/~amathew/notes_thursday.pdf). hopkins and lurie's result admits a generalization to higher heights, so it might be possible to generalize this pd result too. idk how interesting this is though

for theta/delta-rings in particular, there's a lot of good stuff in arxiv.org/pdf/1905.08229.pdf. one consequence of some of the stuff they discuss which i find somewhat interesting is the following: if you take the free E_oo-KU_p-algebra on one generator x, and kill the element psi^p(x) in an E_oo-way, you get an E_oo-KU_p-algebra which has divided powers on x (on homotopy). (basically, you get to now write down x^p/p, and that's basically all you need to get all divided powers on x)

for K(1)-local homotopy in particular, mike hopkins has a fun document called "K(1)-local E∞-ring spectra", which is printed in the tmf book. (as a warning, there's an operation theta which exists in the homotopy of K(n)-local E_n-algebras for any n, and it always satisfies some equation relating theta(x+y) to theta(x) and theta(y). but theta(xy) generally does not satisfy some equation involving theta(x) and theta(y) if the height is >1. idk how relevant this is to the discussion tho)

i think that 2c_6 does map to 2v_1^4 under this map, but this may obviously be wrong

yeah. i guess what i was trying to describe above is actually the map tmf_K(1) -> KO

oy vey, maybe i should shut up and let others give an answer

there is a map tmf -> KO[[q]] given by q-expanding, and i think arun's considering the map tmf -> KO[[q]] -> KO. if what i wrote is correct, then Delta goes to v_1^something, and so this factors through tmf[Delta^-1] = TMF

whoops, yes, thanks dexter. i was wrong earlier, i think c_6 maps to zero

@S.carmeli i think he doesn't have enough reputation to be able to chat in here

@ArunDebray this is a ring map, so i'll say what happens on generators. unit maps to unit (obv), eta^n to eta^n, nothing hits 2v_1^2 (because pi_4 tmf = 0), c_4 goes to v_1^4, and 2c_6 goes to 2v_1^6. i think that's it.

I'd also like to thank the others for being so open with their experiences, it's helpful to hear. part of what's been difficult about this pandemic is that now, more than ever, it's much easier to tie self-worth to productivity (e.g., the memes about newton working out gravity during the plague), motivation for which has dried up a fair amount for me.

@CharlesRezk robert burklund asks me to say: take 2p for p = 1 (mod 3).

this is less algebraic topology than algebraic geometry, but i figure someone here might know an answer. fix r>0. what is an example of a smooth proper variety (necessarily of dimension geq r) over a field k of char p>0 whose hodge-de rham spectral sequence has nontrivial d_r-differential? (i guess this question has a homotopical analogue, too: one can ask for a smooth proper dg-k-linear category C whose tate spectral sequence HH(C)((u)) => HP(C) does not degenerate at E_r.)

this could be a useful description (that you probably already know, and which is related to the kodaira-spencer formalism): Ext^1(G, G_a) = Hom(omega_{G^}, G_a), where omega_{G^} is the invariant differentials in the Cartier dual G^ to G. there's more about this in mazur-messing's "Universal Extensions and One Dimensional Crystalline Cohomology".

the HZ-module associated to Z[S^n] is HZ smash S^n_+ (pointed suspension spectrum), i.e., chains on S^n. so, the HZ-module associated to Z[S^n] (x)_Z Z[S^n] is HZ smash S^n_+ smash S^n_+. But S^n_+ smash S^n_+ = (S^n x S^n)_+, so this is Z[S^n x S^n]

@Dedalus there are many introductions, but here's at least one: arxiv.org/pdf/1907.02904.pdf. it's the chapter by david gepner in haynes' handbook of homotopy theory

@S.carmeli this isn't what you're looking for, but might be interesting anyway: www3.nd.edu/~mbehren1/other/orth.dvi

what about this? define X_n inductively by letting X_0 = sphere spectrum and letting X_n = cofiber of some v_{n-1}-self map of X_{n-1}. there are maps X_n -> X_{n+1}, and the hocolim is dissonant

has anyone got a copy of the videos at math.jhu.edu/~wsw/HOPKINS-MORAVA? i can't seem to download them, and they don't seem to be on the dropbox either

@YuriSulyma in case this helps in searching the literature, there are these things in the world of classical rings called "milnor squares". see e.g. section 1 of arxiv.org/pdf/1612.00418.pdf

thanks for the history!

@TylerLawson ah, i guess that's fair. thanks for clarifying, and apologies for claiming too strong of a statement

i forgot to say *homologically

btw, the reason for the mod 2 homology of any Q_1-algebra already having the dyer-lashof operation Q_1 is simply that the homology of the second space (Q_1)_n = S^1 is exterior on one generator, and that generator gives Q_1

and to answer tim's question, the morava K-theory K(n) is a E_1 (x) Q_1-algebra for p odd

in fact, the universal Q_1-algebra on a sphere in dimension k is the space Ω^2 S^{k+2}, and so this refines the hopkins-mahowald theorem: the free p-local Q_1-algebra with a nullhomotopy of p is HF_p

this thing is awesome: every homotopy commutative ring is a Q_1-algebra (so A_2-algebras are Q_1-algebras), and the mod 2 homology of any Q_1-algebra already has the dyer-lashof operation Q_1 (hence the name)

@TimCampion i'd like to take this discussion as an opportunity to advertise the "cup-1 operad" Q_1, appearing in example 1.3.6 of tyler's contribution to the handbook of homotopy theory. (@TylerLawson, what's the history of this object?)

check out arxiv.org/abs/1902.05046, it's a great summary of the telescope conjecture and other stuff

the instructions in bruner's ext program are confusing

silly tech question: how does one get the adams chart for a finitely presented module over the steenrod algebra?

@EricPeterson thanks, that makes sense. i should've thought about it a little more :p

i know that if i restrict to the obvious copy of S^1 inside SU(n), then the quotient is CP^n, but i don't have a good description of the quotient even if i restrict to the maximal torus of SU(n)

e.g., if n=3, then this is asking about SU(3)\S^7, and if i restrict to the SU(2) action, then SU(2)\S^7 is S^4 by the hopf fibration; but i don't think i know what SU(3)\S^7 is

what is the double coset space SU(n)\SU(n+1)/SU(n) = SU(n)\S^{2n+1}?

@TimCampion they're pretty different: the brown-comenetz dualizing spectrum is not killed by the sphere but is killed by everything in that giant wedge. btw for these sort of questions you should check out "The structure of the Bousfield lattice" by hovey and palmieri

i posted the question at mathoverflow.net/questions/334095/…

ok, will do. thanks @CharlesRezk and @DenisNardin

i was able to prove the statements i wanted which were supposed to be in that document, but i'm guessing there are more interesting results in there

@CharlesRezk i saw it cited in peter may's "Applications and generalizations of the approximation theorem" and in mahowald's "Some homotopy classes generated by ηj". in the former, it is cited as a preprint, but the latter says that it's been "submitted to Springer Lecture Notes Series", and specific sections of that source are even cited (so it must have been fairly complete)