### 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.

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.

#### Keywords

Global injectivity; global invertibility; Hadamard's theorem

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