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.
Topology and its Applications
Brimley, Jacqueline; Griisser, Matthew; Miller, Allison; and Staecker, Christopher P., "The Wecken property for random selfmaps on surfaces with boundary" (2012). Mathematics Faculty Publications. 27.
*Brimley, Jacqueline and Griisser, Matthew and Miller, Allison and Staecker, P. Christopher, The Wecken property for random selfmaps on surfaces with boundary. Topology and its Applications. Accepted for publication 2012. *Jacqueline Brimley is a student co-author. Fairfield University Class of 2013.