On the Nielsen root theory of n-valued maps

Abstract

Let phi :X multimap Y be an n-valued map of connected finite polyhedra and let a in Y. Then, x in X is a root of phi at a if a in phi (x). The Nielsen root number N(phi : a) is a lower bound for the number of roots at a of any n-valued map homotopic to phi . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given epsilon > 0, there is an n-valued map psi homotopic to phi within Hausdorff distance epsilon of phi such that psi has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, q ne 2, then there is an n-valued map homotopic to phi that has N(phi : a) roots at a. We verify the conjecture when X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.