Abstract
Given a continuous map from a 2-dimensional CW complex into a closed surface, the Nielsen root number and the minimal number of roots of satisfy . But, there is a number associated to each Nielsen root class of and an important problem is to know when . In addition to investigate this problem, we determine a relationship between and , when is a lifting of through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.
Highlights
Let f : X → Y be a continuous map between Hausdorff, normal, connected, locally path connected, and semilocally connected spaces, and let a ∈ Y be a given base point
When the range Y of f is a manifold, it is easy to prove that this number is independent of the selected point a ∈ Y, and, from 1, Propositions 2.10 and 2.12, μ f, a is a finite number, providing that X is a finite CW complex
In this case, there is no ambiguity in defining the minimal number of roots of f: μ f : μ f, a for some a ∈ Y
Summary
Let f : X → Y be a continuous map between Hausdorff, normal, connected, locally path connected, and semilocally connected spaces, and let a ∈ Y be a given base point. We define μC f, R to be the minimal cardinality among all Nielsen root classes R , of a map f , H-related to R, for H being a homotopy starting at f and ending at f : Again in 3 was proved that if Y is a manifold, the number μC f, R is independent of the Nielsen root class of f : X → Y. We study this problem for maps from 2-dimensional CW complexes into closed surfaces In this context, we present several examples of maps having liftings through some covering space and not having all Nielsen root classes with minimal cardinality. We present several examples of maps having liftings through some covering space and not having all Nielsen root classes with minimal cardinality Another problem studied in this article is the following. Throughout the text, we simplify write f is a map instead of f is a continuous map
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