[Banach] Abstract of a paper by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic

[Dale Alspach]

This is an announcement for the paper "Sublinear Higson corona and
Lipschitz extensions" by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic.


Abstract: The purpose of the paper is to characterize the dimension
of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz
extensions of functions:
  Theorem: Suppose $(X,d)$ is a proper metric space. The dimension
  of the
sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer
$m\ge 0$ with the following property: Any norm-preserving asymptotically
Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends
to a norm-preserving asymptotically Lipschitz function $g'\colon
X\to \R^{m+1}$.
  One should compare it to the result of Dranishnikov \cite{Dr1}
  who
characterized the dimension of the Higson corona $\nu(X)$ of $X$
is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute
extensor of $X$ in the asymptotic category $\AAA$ (that means any
proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$,
$A$ closed in $X$, extends to a proper asymptotically Lipschitz
function $f'\colon X\to \R^{n+1}$). \par
 In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$
 whose objects
are pointed proper metric spaces $X$ and morphisms are asymptotically
Lipschitz functions $f\colon X\to Y$ such that there are constants
$b,c > 0$ satisfying
 $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$.  We show $\dim(\nu_L(X))\leq
 n$ if and only if $\R^{n+1}$ is an absolute
extensor of $X$ in the category $\tilde\AAA$. \par As an application
we reprove the following result of Dranishnikov and Smith \cite{DRS}:
  Theorem: Suppose $(X,d)$ is a proper metric space of finite
  asymptotic
Assouad-Nagata dimension $\asdim_{AN}(X)$. If $X$ is cocompact and
connected, then $\asdim_{AN}(X)$ equals the dimension of the sublinear
Higson corona $\nu_L(X)$ of $X$.

Archive classification: Metric Geometry; Functional Analysis;
Geometric Topology

Remarks: 13 pages

The source file(s), SublinearHigson.tex: 51559 bytes, is(are) stored
in gzipped form as 0608686.gz with size 15kb. The corresponding
postcript file has gzipped size 76kb.

Submitted from: dydak at math.utk.edu

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