Deformation of maps to branched coverings in dimension three

Abstract

In this paper we address the problem of finding the simplest possible representative within each homotopy class of maps between 2-dimensional manifolds. We shall show that each such surface map of nonzero degree is homotopic to one of an especially simple form. To formulate the main result, define a map f: M -> N between surfaces to be a pinch if there is a compact, connected submanifold M1 c M, with boundary consisting of a single simple closed curve in the interior of M, such that N = M/M1, the quotient of M with Ml identified to a point, and such that f is the quotient map. Define a surjective mapf: M --> N between surfaces to be a branched covering if f is finite-to-one and there exists a finite set B c N, perhaps empty, such that f I f'-(N - B) is an ordinary covering space projection.