  We consider topological conditions under which a locally invertible map
admits a global inverse. Our main theorem states that a local diffeomorphism
$f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the
pre-image of every affine hyperplane is non-empty and acyclic. The proof is
based on some geometric constructions involving foliations and tools from
intersection theory. This topological result generalizes in finite dimensions
the classical analytic theorem of Hadamard-Plastock, including its recent
improvement by Nollet-Xavier. The main theorem also relates to a conjecture of
the aforementioned authors, involving the well known Jacobian Conjecture in
algebraic geometry.
