The classification of maps of nonorientable surfaces

Abstract

A central problem in manifold topology is to classify maps of manifolds up to change of coordinates of domain and range and possibly up to homotopy. In this paper we complete the classification of generic branched coverings of closed surfaces and complete the homotopy classification of maps of positive degree between closed surfaces. Two maps f , g : M , N arc said to be equivalent (resp. equivalent in the homotopy category) if there exist homeomorphisms h : M , M and k : N--,N such that k f= gh (resp. k/'is homotopic to gh). I fk is homotopic to idN we say that f a n d g are strongly equivalent. All maps in this paper preserve basepoints. However discussion of basepoints will be suppressed except where needed. f : M--, N is a branchedeovering if there exists a finite set of points B c N such that f l M f l ( B ) is a covering map. An arbitrary branched covering may be homotoped slightly to be a generic branched covering, i.e. one in which each point of N has d or d 1 preimages where d=degreeJ~ The main results of this paper are