Strong surjectivity of maps from 2-complexes into the 2-sphere
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 \(\delta_{i,j}:=\sum_{\lambda=1}^{r_i}\delta_{(i,j)}^{(\lambda)}\) and define an \(m\times n\) matrix \(\Delta_\mathcal{P}=(\delta_{i,j})\). The authors' main result states that \(f:K_\mathcal{P}\to S^2\) is root free if and only if the diophantine linear system \(\Delta_\mathcal{P}Y=(d_1,\dotsc,d_m)^\top\) has an integer solution.
scientific article; zbMATH DE number 5818006