Vol. 41, No. 2, 1972

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Open mappings on 2-manifolds

William Nathan

Vol. 41 (1972), No. 2, 495–501

This paper will be concerned with the local structure of open, continuous functions f : M2 N1 from a 2-manifold into the real line or the circle. The two main results are:

Theorem 1. Let f : M2 N1 be an open mapping, and suppose that f has isolated branch points. Then for each point p in M2 there are a neighborhood U of p and a positive integer d (depending on p) such that f|U is topologically equivalent to the real analytic mapping z Re(zd). THEOREM 2. If f : M2 N1 is an open, real analytic mapping, then f has isolated branch points, and (hence) the conclusion of Theorem 1 holds.

Mathematical Subject Classification 2000
Primary: 54C10
Secondary: 57A05
Received: 8 April 1970
Revised: 24 September 1971
Published: 1 May 1972
William Nathan