The Wecken property for random maps on surfaces with boundary

TL;DR

This paper investigates the prevalence of the Wecken property among maps on surfaces with boundary, showing that a positive proportion of such maps are Wecken, especially as the fundamental group's rank increases.

Contribution

It provides the first estimates of the asymptotic density of Wecken maps on surfaces with boundary, including explicit bounds related to the fundamental group's rank.

Findings

The density of Wecken maps is nonzero for surfaces with boundary.

Explicit lower bounds for the density are given in terms of the fundamental group's rank.

The bounds approach a limit as the rank of the fundamental group increases.

Abstract

A selfmap is Wecken when the minimal number of fixed points among all maps in its homotopy class is equal to the Nielsen number, a homotopy invariant lower bound on the number of fixed points. All selfmaps are Wecken for manifolds of dimension not equal to 2, but some non-Wecken maps exist on surfaces. We attempt to measure how common the Wecken property is on surfaces with boundary by estimating the proportion of maps which are Wecken, measured by asymptotic density. Intuitively, this is the probability that a randomly chosen homotopy class of maps consists of Wecken maps. We show that this density is nonzero for surfaces with boundary. When the fundamental group of our space is free of rank n, we give nonzero lower bounds for the density of Wecken maps in terms of n, and compute the (nonzero) limit of these bounds as n goes to infinity.