Sublinear Higson corona and Lipschitz extensions

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 $f^{\prime }:A\to {\mathbb{R}}^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g^{\prime }:X\to {\mathbb{R}}^{m+1}$. 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 ${\mathbb{R}}^{n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f:A\to {\mathbb{R}}^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f^{\prime }:X\to {\mathbb{R}}^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f:X\to Y$ such that there are constants $b,c>0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim ({\nu }_L(X))\le n$ if and only if ${\mathbb{R}}^{n+1}$ 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 $X$.

13 pages

References in corpus (1)

• Asymptotic topology