As a refinement of the global invertibility problem, we address the issue of estimating the cardinality of a prescribed fiber \(F^{-1}(q)\) of a locally invertible map solely in terms of objects that are naturally associated to q itself. The following is a prototypical result. Let \(F:\mathbb {R}^n \rightarrow \mathbb {R}^n\) be a local diffeomorphism, \(n\ge 3\) , and \(q\in F(\mathbb {R}^n)\) . We show that q is assumed exactly once by F if the pre-image of every 2-plane containing q, when viewed as a geometric surface in Euclidean n-space, is conformally diffeomorphic to \(\mathbb {R}^2\) . The proofs of this and other theorems involve geometric constructions, the Poincaré-Hopf theorem, the Bôcher theorem on positive harmonic functions, condensers on Riemann surfaces, and elliptic estimates. We conclude with a section that is devoted to invertibility problems related to various aspects of dynamics, algebraic and differential geometry, real and complex analysis. The paper is written in a semi-expository style, as an invitation to global injectivity.
