Abstract:

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, $ngeq 3$, and $qin F(mathbb R^n)$. 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.