Birationality of étale maps via surgery

We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in , then any local diffeomorphism F : X → of simply connected manifolds which is a d -sheeted cover away from D has degree d = 1 or d = ∞ (however all degrees d > 1 are possible if F fails to be a local diffeomorphism at even a single point). In particular, any étale morphism F : X → of algebraic varieties which covers away from such a hypersurface D must be birational.