Subject: Re: question & conf Date: Fri, 14 May 2004 15:03:44 +0100 (BST) From: Ian Leary To: Don Davis CC: 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? > ____________________________________________________ >