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

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 $S_f\cap Z \neq \emptyset$,forany$Z\subset $\mathbb{C}$^n$biregularto$$\mathbb{C}$^{n-1}$and$Z = h^{-1}(0)$forapolynomialsubmersion$h: $\mathbb{C}$^n \to $\mathbb{C}$$. Also, we present topological approaches to the Jacobian conjecture in ${\mathbb{C}}^n$.