Spaces with fibered approximation property in dimension $n$

TL;DR

This paper investigates metric spaces with the fibered approximation property in dimension n, exploring their characteristics and generalizing existing results in the field.

Contribution

It characterizes spaces with the FAP(n) property and extends prior theorems by Uspenskij and Tuncali-Valov.

Findings

Characterization of FAP(n) spaces

Generalizations of Uspenskij's results

Extensions of Tuncali-Valov's theorems

Abstract

A metric space $M$ us said to have the fibered approximation property in dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $\epsilon>0$, $m\geq 0$ and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$ such that $g'$ is $\epsilon$-homotopic to $g$ and $\dim g'\big(\{z\}\times I^n\big)\leq n$ for all $z\in I^m$. The class of spaces having the $\mathrm{FAP}(n)$-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij and Tuncali-Valov.