Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.
@article{bwmeta1.element.doi-10_2478_BF02475955,
author = {Daciberg Gon\c calves and Claudemir Aniz},
title = {The minimizing of the Nielsen root classes},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {112-122},
zbl = {1052.55002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475955}
}
Daciberg Gonçalves; Claudemir Aniz. The minimizing of the Nielsen root classes. Open Mathematics, Tome 2 (2004) pp. 112-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475955/
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