On Topological Approaches to the Jacobian Conjecture in ${\mathbb{C}}^n$

Abstract

We obtain a structure theorem for the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:{\mathbb{C}}^n\to {\mathbb{C}}^n$. Jelonek's results on $S_f$ and our result show that if $f$ is a counterexample to the Jacobian conjecture, then $S_f$ is a hypersurface such that $S_f\cap Z\ne \varnothing$, for any $Z\subset {\mathbb{C}}^n$ biregular to ${\mathbb{C}}^{n-1}$ and $Z=h^{-1}(0)$ for a polynomial submersion $h:{\mathbb{C}}^n\to \mathbb{C}$. Also, we present topological approaches to the Jacobian conjecture in ${\mathbb{C}}^n$.