The classification of maps of nonorientable surfaces
type: Item
instance of: scholarly article
title: The classification of maps of nonorientable surfaces
zbMATH Open document ID: 0656.57001
DOI: 10.1007/BF01456845
author: David Gabai
author: William H. Kazez
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164445
review text: 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.
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 4071921
zbMATH Keywords: classification of generic coverings of closed surfaces
zbMATH Keywords: homotopy classification of maps of positive degree between closed surfaces
zbMATH Keywords: first Stiefel-Whitney class
reviewed by: Vagn Lundsgaard Hansen
MaRDI profile type: Publication
cites work: On the Construction of Branched Coverings of Low-Dimensional Manifolds
cites work: On the classification of generic branched coverings of surfaces
cites work: Deformation of maps to branched coverings in dimension two
cites work: The simple loop conjecture
cites work: The classification of maps of surfaces
cites work: Maps Between Surfaces
cites work: The degree of a map between surfaces
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description: scientific article; zbMATH DE number 4071921