Maps with dimensionally restricted fibers

TL;DR

This paper proves that for certain closed surjective maps between metric spaces with fibers in specific classes, one can find an $F_\sigma$-subset of the domain with controlled dimension properties, extending classical dimension theory results.

Contribution

It introduces a method to find $F_\sigma$-sets within the domain that isolate fibers with zero-dimensional differences, applicable to various classes of spaces including CW-complexes and weakly infinite-dimensional spaces.

Findings

Existence of $F_\sigma$-set $A$ with $ ext{dim} f^{-1}(y)ackslash A=0$ for all $y$

Dimension control of the map $f riangle g$ for generic functions $g$

Extension of classical dimension theory to fibers in classes like CW-complexs and $C$-spaces

Abstract

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.