The classification of maps of nonorientable surfaces (Q1110134)

In this paper the authors complete their interesting work on the classification of generic coverings of closed surfaces and on the homotopy classification of maps of positive degree between closed surfaces. The main results are as follows. Theorem: Two generic branched coverings \(\phi\),\(\psi\) : \(M\to N\) of closed surfaces are strongly equivalent if and only if (i) they have the same degree and (ii) the induced maps on fundamental groups have the same image. Corollary: Two maps f,g: \(M\to N\) of positive degree of closed surfaces are strongly equivalent in the homotopy category if and only if (i) they have the same degree, (ii) the induced maps on fundamental groups have the same image, and (iii) they have the same behaviour on the first Stiefel-Whitney class. Corollary: If f,g: \(G\to H\) are homomorphisms of closed suface groups of equal topological degree greater than zero such that \(f(G)=g(G)\) and the maps of surfaces inducing f and g have the same behaviour on the first Stiefel-Whitney class, then there exists an isomorphism h: \(G\to G\) such that \(f=gh\). The above theorem was proved for the case of N orientable in an earlier paper of the authors [Invent. Math. 90, 219-242 (1987; Zbl 0633.57002)]. The proof of the theorem for the case of N nonorientable is the main task in the present paper.