Subject: Re: question and summer school Date: Thu, 7 Feb 2002 14:06:09 +0100 (MET) From: William Browder Let G be any finite group. Take the m-skeleton of a CW complex model of B_G, and embed it in a large (N-dimensional) euclidean space. A (smooth) regular neighborhood is an m-classifying space, and the boundary has the same homotopuy groups up to dimension about N-m. Since this is a pi-manifold (stably parallelizable) you can do surgery on the interior to make it connected up to the middle dimension. I hope that answers your question, but if you want the manifold to be N dimensional and N classifying at the same time that would imply that its universal covering was a homotopy sphere. Hence G would act freely on a homotopy sphere which is a very strong restriction, and satisfied by very few permutation groups. > Subject: question on BS_n > Date: Wed, 6 Feb 2002 11:38:16 +0300 > From: "Anton Savin" > > I have a question to the algebraic topology discussion list: > > Consider BS_n, where S_n is the permutation group. > Is there an N-classifying manifold model (for N large), denoted by > (BS_n)_N, that is the boundary of a, say, N/2-connected manifold M? > > Remark1. The simplest example I do not understand is n=2. > Remark2. In this problem the bounding manifold is allowed to have > other connected components of the boundary. As an illustration, > one can consider the same problem for (ES_n)_N instead of > (BS_n)_N. This problem is solved by M=finite cyliner with > base (ES_n)_N. > > Anton Savin > _____________________________________________ > >