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

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.