On the non-properness set of a local homeomorphism

We prove that the non-properness set of a local homeomorphism Rⁿ → Rⁿ cannot be ambient homeomorphic to an affine subspace of dimension n − 2. This particularly provides a partial positive answer to a conjecture of Jelonek, that claims the global invertibility of polynomial local diffeomorphisms having non-properness set with codimension greater than 1. Our reasons to obtain this result lead us to some properties of the non-properness set when it has codimension 2, the heart of Jelonek’s conjecture. We also provide a global injectivity theorem related to this conjecture that turns out to generalize previous results of the literature.