Differentiable open maps on manifolds

Abstract

Introduction. This paper contains a detailed discussion with proofs of results announced in [4]. Let Mn and Nn be n-manifolds without boundary, and let f: Mn -+ N n be continuous. The map f is open if, whenever U is open in Mn, f(U) is open in Nn; it is light if, for every y e Nn ,dim (f'(y)) < 0. For n ? 2 there is a canonical light open map Fn,d: E giveni by Fn,d(Xl, x2, ..., Xn) = (u1, u2, x3 *, xv) where Ul + iU2 =(Xl t+ iX2 )d (i =Vi-1; d= 1,2, ... ). For n = 2 it is well known that a nonconstant complex analytic function is open and light. Conversely, Stoilow [12] proved that every light open map is locally topologically equivalent to an analytic map, and thus to some F2,d (d = 1,2, ... ). In fact (1.10), if M2 is compact and f is C2 and open, then f has this canonical structure. The main object of this paper is to prove (2.1) that the corresponding conclusion holds for arbitrary n (n _ 2), if we first remove an exceptional set of dimension at most n 3. Examples are given, especially in ?3, showing that the exceptional set and some of the hypotheses used are necessary. DEFINITION. As in [5] the branch set Bf is the set of points in Mn at which f fails to be a local homeomorphism. NOTATION. Iff : E nEP is C', then fi will be the ith component real-valued function, and Djfi will be the first partial derivative offi with respect to its jth coordinate. If y is a point in En, then yi will be its ith coordinate. The symbols Mn and NP will refer to manifolds of dimensions n and p, respectively. The statement that f: Mn -+ Np is Cm will imply that the manifolds are also Cm. The set of points in Mn at which the Jacobian matrix of f has rank at most q will be denoted by Rq. The closure of a set X is denoted by Cl[X] or X, its interior by int X, and the