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 ?