Preprint A196/2003
Finite branched coverings in a generalized Inverse Mapping Theorem

Carlos Gutierrez | Biasi, Carlos

**Keywords: **
Inversion of maps | injectivity | finite branched coverings

Let $U\subset\R^n$ be an open set and let
$f:U \to \R^n$ be an open continuous map such that,
for all $y\in f(U),$ $f^{-1}(y)$ is a discrete set.
Given $x\in U$, there exist an arbitrarily small
open connected set $V,$ an integer $\ell\ge 1$ and an
$\ell$--fold
covering $f|_V : V\to f(V)$
such that: {\bf (a)} $\overline{V}$ and $f(\overline{V})$ are
neighborhoods of $x$ and $f(x)$, respectively, {\bf (b) }
$f|_{\overline{V}}:\overline{V} \to f(\overline{V})$ is a proper map,
and {\bf (c) } for all $y\in f(U),$
$\# (f^{-1}(y) \cap \overline{V}) \le \ell.$
Moreover, if $|\deg(f, x)|\equiv 1,$
then $f$ is locally homeomorphic. We discuss conditions under
which locally homeomorphic maps of a manifold are global homeomorphisms.