On finite-dimensional maps and other maps with "small" fibers
We prove that if f is a $k$-dimensional map on a compact metrizable space $X$ then there exists a σ-compact $(k-1)$-dimensional subset $A$ of $X$ such that $f|X∖A$ is 1-dimensional. Equivalently, there exists a map $g$ of $X$ in $I^k$ such that $\dim(f × g)=1$. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that $f(X)$ is finite-dimensional. These results are then extended to maps with fibers restricted to some classes of spaces other than the class of $k$-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then $\dim(f|X∖A) ≤ 1$ for some σ-compact weakly infinite-dimensional subset $A$ of $X$. The proof applies essentially the properties of hereditarily indecomposable continua.