A new construction of universal spaces for asymptotic dimension

TL;DR

This paper constructs universal metric spaces $mbda_n$ for each dimension $n$, serving as models for all spaces with asymptotic or uniform dimension at most $n$, unifying large-scale and small-scale topology.

Contribution

It introduces a novel construction of universal spaces that simultaneously capture asymptotic and uniform dimensions for separable metric spaces.

Findings

Universal spaces $mbda_n$ have dimension exactly $n$.

Any space with dimension ≤ $n$ embeds into $mbda_n$ in the respective category.

The spaces unify large-scale and small-scale topological properties.

Abstract

For each $n$, we construct a separable metric space $\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\mathop{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\mathop{udim}$) at most $n$. Thus, $\mathbb{U}_n$ serves as a universal space for dimension $n$ in both the large-scale and infinitesimal topology. More precisely, we prove: \[ \mathop{asdim} \mathbb{U}_n = \mathop{udim} \mathbb{U}_n = n \] and such that for each separable metric space $X$, a) if $\mathop{asdim} X \leq n$, then $X$ is coarsely equivalent to a subset of $\mathbb{U}_n$; b) if $\mathop{udim} X \leq n$, then $X$ is uniformly homeomorphic to a subset of $\mathbb{U}_n$.