Subject: response on BS_n Date: Fri, 15 Feb 2002 12:31:21 +0300 From: "Anton Savin" To: Dear Professor Davis, This letter is a reply to the message posted by Professor Browder. Anton Savin _______________________________________________________ 1. I would like to thank Professor Browder and Professor Schick for their answers to my question. Their method solves a geometric problem that I would like to explain. 2. Consider compact smooth manifolds M with boundary dM represented as a finite covering pi:dM --> X over a smooth X with n leafs. For the triples of the form (M,X,pi) there is a notion of embedding (M,X,pi) in (N,Y,pi'). This is an embedding f:M->N of manifolds with boundary that commutes with the projections and induces ÉÛÏÕÓÅÛÝÔÙ of the fibers of pi and pi'. In this category we would like to find a universal space for embeddings. The term universal means that for an (M,X,pi) there should be a unique embedding (up to isotopy) in the universal space. Remark. For a related problem, where we consider only coverings corresponding to principal G-bundles (G-finite group) the corresponding universal triple is ([0,1)*EG, BG, pi_N), where (BG) is an N-classifying space for principal G-bundles (we omit N for brevity). In the general case, for pairs (dM,X) the universal pair for embeddings is (BS_{n-1}, BS_n). Thus, if BS_{n-1}is represented as a boundary of a highly connected manifold M_0 (this is shown in the letter by Prof. Browder) then these properties make it possible to show that M_0 is a universal space for embeddings of (M,X,pi) (of course with dim M bounded). 3. I would like to conclude this letter with a refinement of my question. In the group case (see Remark 1) the projection pi is naturally defined on the entire manifold with boundary. Is it possible to find a similar model with the projection defined everywhere in the general case ?