Subject: Re: response and conf Date: Fri, 21 May 2004 14:22:00 -0500 (CDT) From: Peter Webb As an addendum to Ian Leary's response to Jack Morava's question about (BG)/G where G acts by conjugation, let me mention that about 5 years ago I calculated the second mod 2 homology of this space when G is the quaternion group of order 8. It came out to have dimension 3, whereas the second mod 2 homology of BG has dimension 2. In doing this I was motivated by trying to come up with a construction which would give a classifying space for a block of G, and was simply experimenting. I did this calculation by computer (using a package I wrote in GAP which computes the homology of nerves of categories - see http://www.math.umn.edu/~webb), and I did it not with (BG)/G but with a space which is a certain homotopy colimit which I now describe. It is the homotopy colimit of a functor defined on the category whose objects are the subgroups of G, and where the morphisms are the group homomorphisms which have the form conjugation by an element of G followed by inclusion. The functor takes each subgroup P to that subgroup regarded as a category, and then we follow by taking the nerve of the category to get a copy of BP. Each group homomorphism P -> Q may be regarded as a functor between these categories, and hence we get a morphism of the nerves BP -> BQ thus giving the functorial dependence. Conjugation by elements of the centralizer of P thus gives rise to trivial morphisms, but conjugation by the elements of P need not. (I mention this, because there is another construction of BP giving a different functorial dependence.) The advantage for me of working with the homotopy colimit was that by Thomason's theorem it can be expressed as the nerve of the Grothendieck construction, and hence my software was applicable. I wrote to several people with the result of my calculation to ask them if it seemed plausible. Bill Dwyer wrote back to me with arguments which use various Grothendieck constructions to show that the homotopy colimit I was working with is in fact (BG)/G. Regards, Peter Webb On Fri, 14 May 2004, Don Davis wrote: > Two postings: A response and a conf update..........DMD > ______________________________________________________ > > Subject: Re: question & conf > Date: Fri, 14 May 2004 15:03:44 +0100 (BST) > From: Ian Leary > > For a discrete group G, the map BG ---> BG/G > is a homotopy equivalence if and only if G is > abelian, in which case G acts trivially on BG. > > The fundamental group of BG/G is the abelianization > of G. > > There is a way to think about this in terms of classifying > spaces for families of subgroups. Let G x G act on G by: > > (g,h).k = gkh^{-1}. > > This action is transitive and the stabilizer of the > identity element is the diagonal subgroup of elements > of the form (g,g) in G x G. > > Now take the usual construction of EG, as the simplicial > set with n-simplices the set G^{n+1}, with the given > action of G x G. A subgroup H of G x G fixes some point > of EG if and only if H is conjugate to a subgroup of the > diagonal copy of G. Furthermore, the fixed point set > for any such subgroup is contractible. It follows that > EG with this action of G x G is a classifying space for > a family F of subgroups of G x G. A group H is in > the family F if and only if H is conjugate to a subgroup > of the diagonal subgroup. > > Factoring out EG by the action of G x 1 gives BG, > and then factoring out by the rest of G x G gives BG/G. > This gives a way to see that the fundamental group of > BG/G is the abelianisation of G. I don't know what the > meaning of the higher homotopy groups of BG/G is. > > One could ask what homotopy types can occur as BG/G > for some discrete G, a sort of Kan-Thurston type question. > > Best wishes, > Ian Leary > > On Thu, 13 May 2004, Don Davis wrote: > > > Two postings: A question and a conference........DMD > > _____________________________________________________ > > > > Subject: question for the list > > Date: Thu, 13 May 2004 08:56:51 -0400 (EDT) > > From: Jack Morava > > > > Here's a proposed question for the list. > > > > A (finite) group G acts on itself through conjugation, > > so its classifying space BG inherits a G-action. It's > > easy to see that the action of any element g on BG > > is homotopic to the identity. > > > > I've always assumed that the quotient map > > > > BG --> BG/G > > > > is a homotopy equivalence, but this doesn't follow > > from the remark above. [If C is a connected topological > > group acting on a space X then the action of any > > element of C is homotopic to the identity, but that > > doesn't imply that X --> X/C is an equivalence!] > > > > Is this in fact true, well-known, false...? Is > > there a standard (or accesible) reference for > > it either way? >