Representing homotopy classes by maps with certain minimality root properties. II (Q2236454)

Let \(M\) be a surface in \(S^2\vee S^1\), \(A\in M\), \(B\in S^2\), \(P=S^1\cap S^2\) and \(D_A, D_B\) disks centered in \(A\) and \(B\), respectively. In this paper the authors show that for \(f:M\to S^2\vee S^1\) it is possible to find \(h:M\to S^2\vee S^1\) and \(k:S^2\to S^2\vee S^1\) such that \(h\) can be homotopically factored by \(h^{'}:M\to S^1\), that is, if \(\iota:S^1\to S^2\vee S^1\) is the inclusion then \(\iota\circ h^{' }=h\), \(h(A)=k(B)\) and \(f\simeq H(h,k)\) defined by \[ H(h,k)=\left\{ \begin{array}{c} h(z), z\in M- \mathring{D}_A\\ k(z), z\in S^2- \mathring{D}_B) \end{array}\right. .\] Then supposing that the decomposition \(\left[f\right]=\left[H(h,k)\right]\) is such that \(h\) and \(k \) are not null homotopic, the authors show that \begin{itemize} \item[1.] For \(y\in S^2-P\), there exists \(k_1\in [k]\) such that \(k_1^{-1}(y)\) is the root set for \(f\) since \(y \notin\mbox{image}(h)\). \item[2.] For \(y\in S^1\), there exists \(h_1\in [h]\) and \(k_2\in [k]\) such that \(h_1^{-1}(y)\cup k_2^{-1}(y)\) is the solution of the root problem for \(f\). \end{itemize} For Part I of this paper see [\textit{N. C. L. Penteado} and \textit{O. M. Neto}, Proc. R. Soc. Edinb., Sect. A, Math. 146, No. 5, 1005--1015 (2016; Zbl 1360.55001)].