A new construction of universal spaces for asymptotic dimension

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 ($\mathrm{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\mathrm{udim}$) 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 $\mathrm{asdim}X\le n$, then $X$ is coarsely equivalent to a subset of ${\mathbb{U}}_n$; b) if $\mathrm{udim}X\le n$, then $X$ is uniformly homeomorphic to a subset of ${\mathbb{U}}_n$.

References in corpus (1)

• Characterizing Nobeling spaces