Strong surjectivity of maps from 2-complexes into the 2-sphere
type: Item
instance of: scholarly article
title: Strong surjectivity of maps from 2-complexes into the 2-sphere
zbMATH Open document ID: 1207.55002
author: Marcio Colombo Fenille
author: Oziride Manzoli Neto
published in: Central European Journal of Mathematics
publication date: 2010-11-22T00:00:00Z
review text: Let \(f:K\to S^2\) be a map from a 2-complex into the 2-sphere, denote by \(\mu(f)\) the minimal cardinality of \(\phi^{-1}(\{a\})\) where \(\phi\) is homotopic to \(f\) (here, of course, \(a\in S^2\) and the definition is independent from this choice). If \(\mu(f)>0\) the map \(f\) is said to be strongly surjective (or, equivalently, if the Nielsen number \(N(f)\not=0\)). On the other hand, if \(\mu(f)=0\) the map \(f\) is said to be root free. In the present article the authors describe necessary and sufficient conditions in terms of linear Diophantine systems for \(f\) to be root free. In order to give a more detailed description, let \(f\) be cellular and consider the minimal cellular decomposition of \(S^2\) as \(e^0_*\cup e^2_*\) with one 0-cell and one 2-cell. Suppose that \(K\) has \(m\) 2-cells, \(e_1^2,\dotsc,e_m^2\) and let \(K^1\) be the 1-skeleton of \(K\). Collapse \(K^1\) to a point \(e^0\) and denote by \(\omega:K\to K/K^1\) the corresponding quotient map. Then \(f\) factors in a unique way over \(K/K^1\) as \(f=\bar{f}\circ\omega\). Fix a generator \(\rho\) of \(H^2S^2\) and choose generators \(\rho_1,\dotsc,\rho_m\) of the free abelian group \(H^2(K/K^1)\). Then there are integers \(d_1,\dotsc,d_m\) such that \(\bar{f}^*(\rho)=d_1\rho_1+\dotsb+d_m\rho_m\). Let \(\mathcal{K}_\mathcal{P}\) the model 2-complex of the group presentation \(\mathcal{P}=\left<x_1,\dotsc,x_n|\;\mathbf{r}_1,\dotsc,\mathbf{r}_m\right>\) [cf., \textit{A. J. Sieradski}, Algebraic topology for two-dimensional complexes.Two-dimensional homotopy and combinatorial group theory. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 197, 51-96, 381-407 (1993; Zbl 0811.57002)]. For \(i=1,\dotsc,m\) write the relations of the group presentation \(\mathcal{P}\) as \(\mathbf{r}_i=\left(x_1^{\delta_{i,1}^{(1)}}\dotsm x_n^{\delta_{i,n}^{(1)}}\right)\dotsm\left(x_1^{\delta_{i,1}^{(r_i)}}\dotsm x_n^{\delta_{i,n}^{(r_i)}}\right)\). Let then \(\delta_{i,j}:=\sum_{\lambda=1}^…
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57M20
zbMATH DE Number: 5818006
zbMATH Keywords: strong surjectivity
zbMATH Keywords: two-dimensional complex
zbMATH Keywords: group presentation
zbMATH Keywords: Diophantine linear system
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: s11533-010-0031-6
OpenAlex ID: W2063934310
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DOI: 10.2478/S11533-010-0031-6
description: scientific article; zbMATH DE number 5818006