htt now has hyperlinks!

yesss

are the Ravenel X(n) E_oo-ring spectra?

they're E_2-ring spectra

They are thom spectra of maps of loop spaces, but I don't think those guys are infinite loop spaces. That would imply what you are asking, if the map was an infinite loop map, but isn't necessary.

(at p=2, my knowledge at odd primes is below remedial)

is it because one of the maps of loop spaces defining X(n) isn't a 3-fold loop space map?

well, kind of. that's enough to show you that you don't automatically get an E_3-structure, but sometimes that's not an if and only if. e.g. the construction of HZ/p as a Thom spectrum is only by a map of 2-fold loop spaces, and so it only gets an E_2-structure from that even though it has an E_infty structure

the proof goes something like: if X(n) was E_3, then the map X(n) -> HZ/2 could be made an E_3-map; then the map from H_* X(n) to the dual Steenrod algebra would be a map of algebras with E_3-Dyer-Lashof operations; but the image H_* X(n) is not closed in the dual Steenrod algebra under the E_3-Dyer-Lashof operations

@TylerLawson I somehow don't believe this.

i think I wrote somebody an email about this a while ago that's a little more expanded, I can try and dig it up

@SeanTilson I seriously can't remember even what degrees you can apply the \beta P^i most of the time, much less remember Adem relations or anything

Yes, but you don't assume it is the same, right?

I mean, you used the word remedial.

i'm speaking truth which has been slightly exaggerated for effect

Oh Tyler, you are so funny.

@TylerLawson to be fair there are something like 3 different indexing conventions on power operations in homology. I get confused anew each time...

Does the category of abelian torsion groups have a generator? It has a cogenerator, but I'm unsure about a generator

@TylerLawson oh, ok; which Dyer-Lashof operation produces a contradiction? (maybe you could give a reference)

@skd I don't know which one is the first to actually cause a problem (it depends on n).

But H_* X(n) is a polynomial algebra on finitely many generators, and its image contains the generator xi_1^2 in degree 2 of the dual Steenrod algebra. The infinite polynomail elements xi_n^2 are given by E-3-Dyer-Lashof operations on xi_1^2 by work of Steinberger (this is in the H_infty book).

roughly Q^4 xi_1^2 = x_2^2, Q^8 xi_2^2 = xi_3^2, etc (possibly mod decomposables / using conjugate classes)

And so H_* X(n), because it's a poly. algebra on finitely many variables, isn't big enough to be surjective onto this polynomial algebra in infinitely many variables that is generated by xi_1^2

oh, I see

that's pretty neat