Copyright © 2006 Robert F. Brown. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f:X→X be a map of a compact, connected Riemannian manifold, with or without boundary. For ∈>0 sufficiently small, we introduce an ∈-Nielsen number N∈(f) that is a lower bound for the number of fixed points of all self-maps of X that are ∈-homotopic to f. We prove that there is always a map g:X→X that is ∈-homotopic to f such that g has exactly N∈(f) fixed points. We describe procedures for calculating N∈(f) for maps of 1-manifolds.