Foliations and Global Inversion

TL;DR

This paper establishes topological conditions involving foliations under which a local diffeomorphism from a manifold to Euclidean space is globally invertible, extending classical results and relating to the Jacobian Conjecture.

Contribution

It provides a new topological criterion for global invertibility of local diffeomorphisms using foliation and intersection theory, generalizing classical theorems.

Findings

A local diffeomorphism is bijective if certain homology and acyclicity conditions are met.

The proof employs geometric constructions involving foliations and intersection theory.

Results relate to the Jacobian Conjecture in algebraic geometry.

Abstract

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