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stub for Barratt-Eccles operad
I had reason to spell out at Barratt-Eccles operad the structure, as a simplicial operad, in full explicit detail. So I did.
I corrected some spelling typos i.e. Barrat instead of Barratt. :-)
Thanks. I did that pretty consistently, didn’t I?
Since you introduced links to their names: what’s their full names? I can’t find them on Wikipedia, and I didn’t find their original article online (and even if I did, it may not contain their full names).
Michael Barratt and Peter Eccles, but I have already added entries for them.
Thanks!
Tim, while we are at it: do you happen to know some link that we could supply in an entry on Chris Reedy (as in Reedy model category)?
I cannot find one. There is a math genealogy page (http://genealogy.math.ndsu.nodak.edu/id.php?id=43120), which I think is him. There is a possible computer scientist with that name at WWU! but as I cannot recall meeting him I cannot tell if it is the same person.
Thanks. That must be him. I created a page here.
What would be an explicit (as combinatorial as possible) description (in topological world) of the universal bundle of the symmetric group $\Sigma_n$, used for Barratt-Eccles operad ?
I improved and added some references.
What would be an explicit (as combinatorial as possible) description (in topological world) of the universal bundle of the symmetric group $\Sigma_n$,
How about this: let $\Sigma_n // \Sigma_n$ be the action groupoid of $\Sigma_n$ acting on itself, and $*// \Sigma_n$ that of its trivial action on the point. Write $\Sigma_n // \Sigma_n \to *//\Sigma_n$ for the canonical functor.
A model for the universal $\Sigma_n$-bundle in topological spaces is the geometric realization of the nerve of this functor.
Or could we embed $\Sigma_n$ into $GL_n$ as permutation matrices and let it act on the infinite Stiefel manifold? I have no idea what the quotient will be, but the total space is ’quite combinatorial’ (Schubert cells and so forth)
Surely, Urs, this construction could lead eventually to a clean combinatorial description, but as it stands, it is just an abstract formulation valid for any discrete group in place of $\Sigma_n$. I really need a non-repeating bookkeeping of $k$-cells, how they fit, and the action and then how these data fit when also $n$ grows (to fit various $E \Sigma_n$). I need eventually to play with certain tensor product for algebras over $E \Sigma_*$-operad, so I should calculate with cells appearing the nerves you mention. Probably somebody has written a clean combinatorial description somewhere, which could be directly used.
P.S. the question is related to my attempt to undestand better the setup of Grisha’s technical problem pdf which is eventually motivated by some hard computations in algebraic K-theory; and where the data are conveniently expressed with this particular model of $E_\infty$-operad.
David: this model may be probably useful once I solve the main problem. Good idea.
I don’t understand. This seems to be the most explicit combinatorial description that one could hope for. Using this you can easily write down all cells at any degree explicitly on a small piece of paper.
As I clearly stated, it is only as explicit as for any other finite group, i.e. just in terms of abstract elements of the original group and the equality (and in those terms counting depends on phenomena which should be tested element by element and is not determined a priori). Unless some extra genuine work is done with generators and relations we do not have more than that, and the latter is what I was asking for. You may be right that this indeed may as well be enough for my immediate purposes, but truly specific combinatorial description for the symmetric group must involve the generators, i.e. the elementary transpositions. This is important to determined which cells (and which results of boundary maps) are equal and which are not (word problem). In nerve description one has to do concatenations when calculating boundary maps (for instance). Now when you concatenate one has reordering phenomena reducing the length in terms of generators. If we do not do that than we do not distinguish different words and if we don’t we may count smaller connectedness for various subcomplexes. So one needs to have a unique description of any word and the only way I know this for symmetric group is the standard one with elementary transpositions as in the theory of Coxeter groups. People who compute the finite group cohomology (notoriously involved combinatorial task for given finite group), hence need classifying spaces, use such things in computations, so I thought there is some standard description prepared for that.
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