Lift factors for the Nielsen root theory on $n$-valued maps
DOI:
https://doi.org/10.12775/TMNA.2022.017Keywords
55M20Abstract
A root of an $n$-valued map $\varphi \colon X \to D_n(Y)$ at $a \in Y$ is a point $x \in X$ such that $a \in \varphi(x)$. We lift the map $\varphi$ to a split $n$-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of $\varphi$. We describe the relationship between the root classes at $a$ of the lift factors and those of $\varphi$. We define the Reidemeister root number $\RR (\varphi)$ in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number $NR(\varphi)$, the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the $\Phi$-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if $Y$ is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of $\varphi$. If $X$ and $Y$ are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of $\varphi$ that is invariant under $\Phi$-relation and this implies that if its index is non-zero, then the root class is essential.References
R. Brooks, Certain subgroups of the fundamental group and the number of roots of f (x) = a, Amer. J. Math. 95 (1973), 720–728.
R. Brooks, Nielsen Root Theory, Handbook of Topological Fixed Point Theory, Springer, 2005, pp. 375–431.
R. Brown, Fixed points of n-valued multimaps of the circle, Bull. Polish Acad. Sci. 54 (2006), 153–162.
R. Brown, Nielsen numbers of n-valued fiber maps, J. Fixed Point Theory Appl. 4 (2008), 183–201.
R. Brown, On the Nielsen root theoryof n-valued maps, J. Fixed Point Theory Appl. 23 (2021), no. 4, paper no. 51, 10 pp.
R. Brown, C. Deconinck, K. Dekimpe and P.C. Staecker, Lifting classes for the fixed point theory of n-valued maps, Topology Appl. 274 (2020), 26 pp.
R. Brown and D. Gonçalves, On the topology of n-valued maps, Adv. Fixed Point Theory 8 (2018), 205–220.
R. Brown and K. Kolahi, Nielsen coincidence, fixed point and root theories of n-valued maps, J. Fixed Point Theory Appl. 14 (2013), 309–324.
V. Hansen, Braids and coverings: selected topics, London Math. Soc. Student Texts 18, (1989).
A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
H. Hopf, Zur Topologie der Abbildungen von Manigfaltikeiten II, Math. Ann. 102 (1930), 562–623.
W. Massey, Algebraic Topology: An Introduction, Harcourt, Brace, 1967.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Robert F. Brown, Daciberg Lima Gonçalves
![Creative Commons License](http://i.creativecommons.org/l/by-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0