Foliations and global inversion

Abstract

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f:M\to R^n$ 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.