  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 $S_f\cap Z \neq
\emptyset$, 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
$\mathbb{C}^n$.
