2021/06/03 by Dinh, Si Tiep, Pham, Tien Son
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.01593
For a definable continuous mapping f from a definable connected open subset Ω of \mathbb Rn into \mathbb Rn, we show that the following statements are equivalent: (i) The mapping f is open. (ii) The fibers of f are finite and the Jacobian of f does not change sign on the set of points at which f is differentiable. (iii) The fibers of f are finite and the set of points at which f is not a local homeomorphism has dimension at most n - 2. As an application, we prove that Whyburn's conjecture is true for definable mappings: A definable open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism.