Incompressibility and global inversion

Abstract

Given a local diffeomorphism $f\colon \mathbb{R}^n\to \mathbb{R}^n$, we consider certain incompressibility conditions on the parallelepiped $Df(x)([0,1]^n)$ which imply that the pre-image of an affine subspace is non-empty and has trivial homotopy groups. These conditions are then used to establish criteria for $f$ to be globally invertible, generalizing in all dimensions the previous results of M. Sabatini.