Mathematical Research Letters

Volume 24 (2017)

Number 4

The asymptotic density of Wecken maps on surfaces with boundary

Pages: 1133 – 1145

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n4.a9

Authors

Seung Won Kim (Department of Mathematics and Applied Statistics, Kyungsung University, Busan, South Korea)

P. Christopher Staecker (Department of Mathematics, Fairfield University, Fairfiel, Connecticut, U.S.A.)

Abstract

The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed points among maps homotopic to $f$. When these numbers are equal, the map is called Wecken. The paper by Brimley, Griisser, Miller, and the second author investigates the abundance of Wecken maps on surfaces with boundary, and shows that the set of Wecken maps has nonzero asymptotic density.

We extend the previous results as follows: When the fundamental group is free with rank $n$, we give a lower bound on the density of the Wecken maps which depends on $n$. This lower bound improves on the bounds given in the previous paper, and approaches $1$ as $n$ increases. Thus the proportion of Wecken maps approaches $1$ for large $n$. In this sense (for large $n$) the known examples of non-Wecken maps represent exceptional, rather than typical, behavior for maps on surfaces with boundary.

Keywords

fixed points, surface, asymptotic density, Nielsen theory

2010 Mathematics Subject Classification

37C20, 55M20

Received 3 May 2012

Accepted 23 November 2015

Published 9 November 2017