53 research outputs found
Homotopy type of the complement of an immersion and classification of embeddings of tori
This paper is devoted to the classification of embeddings of higher
dimensional manifolds. We study the case of embeddings ,
which we call knotted tori. The set of knotted tori in the the space of
sufficiently high dimension, namely in the metastable range ,
, which is a natural limit for the classical methods of embedding
theory, has been explicitely described earlier. The aim of this note is to
present an approach which allows for results in lower dimension
Sublinear Higson corona and Lipschitz extensions
The purpose of the paper is to characterize the dimension of sublinear Higson
corona of in terms of Lipschitz extensions of functions:
Theorem: Suppose is a proper metric space. The dimension of the
sublinear Higson corona of is the smallest integer with
the following property: Any norm-preserving asymptotically Lipschitz function
, , extends to a norm-preserving
asymptotically Lipschitz function .
One should compare it to the result of Dranishnikov \cite{Dr1} who
characterized the dimension of the Higson corona of is the
smallest integer such that is an absolute extensor of
in the asymptotic category \AAA (that means any proper asymptotically
Lipschitz function , closed in , extends to a
proper asymptotically Lipschitz function ). \par
In \cite{Dr1} Dranishnikov introduced the category \tilde \AAA whose
objects are pointed proper metric spaces and morphisms are asymptotically
Lipschitz functions such that there are constants
satisfying
for all .
We show if and only if is an absolute
extensor of in the category \tilde\AAA. \par As an application we reprove
the following result of Dranishnikov and Smith \cite{DRS}:
Theorem: Suppose is a proper metric space of finite asymptotic
Assouad-Nagata dimension \asdim_{AN}(X). If is cocompact and connected,
then \asdim_{AN}(X) equals the dimension of the sublinear Higson corona
of .Comment: 13 page
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