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Article overview
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The Wecken property for random maps on surfaces with boundary | Jacqueline Brimley
; Matthew Griisser
; Allison Miller
; P. Christopher Staecker
; | Date: |
1 Sep 2011 | 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. | Source: | arXiv, 1109.0218 | Services: | Forum | Review | PDF | Favorites |
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