Geometric aspects of global invertibility [ Back ]

12:00 to 13:00
UAB - Dept. Matemàtiques (C1/-128)
Fred Xavier
Texas Christian University and University of Notre Dame


We address the problem of estimating the cardinality of a prescribed fiber $F^{-1}(q)$ of a locally invertible map in terms of objects naturally associated to $q$ itself. Let $F:\mathbb R^n \to \mathbb R^n$ be a local diffeomorphism, $n\geq 3$, and $q\in F(\mathbb R^n)$. Using topological, geometric, and analytic arguments we show that if the pre-image of every plane containing $q$ is a Riemannian submanifold of Euclidean space that is conformally diffeomorphic to $\mathbb R^2$, then $q$ is assumed exactly once by $F$. This and other results are special cases of a general abstract global invertibility theorem that also yields necessary and sufficient conditions for invertibility in the Jacobian conjecture (JC). Motivated by these results, a natural conjecture in approximation theory is formulated whose validity implies that (JC)  does not hold, without the need to exhibit a counterexample.