Minimal Nielsen root classes and roots of liftings
Let \(f: X\to Y\) be a map, and \(a\) be a given point in \(Y\). A root of \(f\) at \(a\) is a point \(x\) in \(X\) such that \(f(x)=a\). In Nielsen root theory, one is interested in determining the number \(\min _{g\simeq f}\sharp g^{-1}(a)\), where \(g\) ranges over the maps homotopic to \(f\). If the target space \(Y\) is a closed manifold, this number is independent of the choice of the base point \(a\), denoted \(\mu(f)\). By using traditional methods in Nielsen theory, the root set of a map splits into a disjoint union of root classes. There is a well-defined one-to-one correspondence between the sets of root classes of two homotopic maps. A root class is said to be essential if it is always non-empty under any correspondence determined by a homotopy. The Nielsen root number \(N(f,a)\) is defined to be the number of essential root classes. Clearly, \(N(f,a)\) gives a lower bound of \(\mu(f)\). Given a root class \(\mathcal{R}\) of \(f\), \(\mu_C(f, \mathcal{R})\) is defined to be the minimal number \(\min_{g\simeq f} \sharp \mathcal{R}_{g}\), where \(g\) ranges over the maps homotopic to \(f\) and \(\mathcal{R}_{g}\) is the root class of \(g\) homotopically corresponding to the given root class \(\mathcal{R}\). It is obvious that the invariants mentioned above satisfy the relation \(N(f,a)\leq\sum_{\mathcal{R}} \mu_C(f, \mathcal{R})\leq \mu(f)\). A natural question is: under which conditions do these inequalities become equalities? \textit{D. Gonçalves} and \textit{C. Aniz} proved in [Cent. Eur. J. Math. 2, No.~1, 112--122 (2004; Zbl 1052.55002)] that \(\mu_C(f, \mathcal{R})\) is independent of the choice of the root class \(\mathcal{R}\) of a given map \(f: X\to Y\) if \(Y\) is a manifold. In this case, one may simply write \(\mu_C(f)\). Hence, one may ask for a condition which ensures that \(N(f)\mu_C(f) =\mu(f)\). It is equivalent to ask for sufficient conditions to minimize simultaneously the root numbers in all root classes. By using some examples, the authors of the paper under review show that \(\mu(f)\) can be arbitrarily larger than \(N(f)\mu_C(f)\) for maps from a two-dimensional complex to the torus. When the target manifold is the projective plane \(\mathbb RP^2\), some sufficient conditions are obtained. Moreover, the relation between minimal root number \(\mu(f)\) of a map and those \(\mu(\tilde f)\)'s of its lifts with respect to a \(k\)-sheeted covering is also considered, showing that \(\mu(f)\geq k \mu(\tilde f)\).
scientific article; zbMATH DE number 5627843