by  S.Yu. Orevkov

Abstract:   We prove the following result. There exists a complex analytic 2-manifold   X   with the boundary   dX,   a smooth analytic disk   M   embedded into   X,   transversal to   dX,   with   dM   on   dX,   and a holomorphic three-sheeted branching covering   f : X ---> B^4,   where   B^4   is the unit ball in   C²,   such that   Int X - M   is homeomorphic to   R^4,   the restriction of   f   onto   M   is an embedding, and   f   has branching of order two along   M,   being an immersion (i.e. local homeomorphism) everywhere on   Int X - M.  

Without analytic structure, this example was constructed by A.G.Vitushkin. It was motivated by the well-known Jacobian Conjecture. We give also an alternative proof of this Vitushkin's result. The analytic realization is based on a construction due to Lee Rudolph.