Nollet–Xavier hyperplane-preimage injectivity conjecture

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a local diffeomorphism, meaning a smooth map that is locally invertible. An affine hyperplane in $\mathbb{R}^n$ is a translate of a codimension-one linear subspace.

Nollet–Xavier conjecture. If the pre-image $f^{-1}(H)$ of every affine hyperplane $H\subseteq\mathbb{R}^n$ is connected, possibly empty, then $f$ is injective.

This conjecture links the connectedness of level sets of a locally invertible map to global injectivity. It remains open and would imply the Jacobian Conjecture in algebraic geometry, which asserts that every polynomial local biholomorphism $F:\mathbb{C}^n\to\mathbb{C}^n$ is invertible.