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
> _____________________________________________
>
>