Minimal number of preimages under maps of surfaces
type: Item
instance of: scholarly article
title: Minimal number of preimages under maps of surfaces
zbMATH Open document ID: 1112.55001
author: E. A. Kudryavtseva
author: Heiner Zieschang
author: S. A. Bogatyi
author: Daciberg Lima Gonçalves
published in: Mathematical Notes
publication date: 2007-03-09T00:00:00Z
review text: The authors consider a map \(f:M_1\to M_2\) of closed surfaces and \(c\in M_2\). Call \(A(f)\) the absolute degree of \(f\) (cf., [\textit{D. B. A. Epstein}, Proc. Lond. Math. Soc. 16, 369--383 (1966; Zbl 0148.43103)]) and denote by \(\ell(f)\) the index of \(f_\#(\pi_1(M_1))\) in \(\pi_1(M_2)\). Let \(MR(f):=\min| \bar{f}^{-1}(\{c\})| \) where the minimum is taken over all \(\bar{f}\) which are homotopy equivalent to \(f\). Similarly let \(NR(f)\) denote the Nielsen coincidence number for \(f\) and the constant mapping \(c\). The authors prove that \(MR(f)=NR(f)=0\) provided \(A(f)=0\). If \(A(f)>0\) one gets \(MR(f)=\max\{\ell(f),\chi(M_1)+(1-\chi(M_2))A(f)\}\) and \(NR(f)=\ell(f)\).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 5132495
zbMATH Keywords: closed surface
zbMATH Keywords: root problem
zbMATH Keywords: Nielsen coincidence number
zbMATH Keywords: Wecken property
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: b:matn.0000015017.47636.4b
OpenAlex ID: W2023836722
DOI: 10.1023/B:MATN.0000015017.47636.4B
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description: scientific article; zbMATH DE number 5132495