Nielsen coincidence, fixed point and root theories of n-valued maps

Abstract

Let $${(\phi, \psi)}$$ be an (m, n)-valued pair of maps $${\phi, \psi : X \multimap Y}$$ , where $${\phi}$$ is an m-valued map and $${\psi}$$ is n-valued, on connected finite polyhedra. A point $${x \in X}$$ is a coincidence point of $${\phi}$$ and $${\psi}$$ if $${\phi(x) \cap \psi(x) \neq \emptyset}$$ . We define a Nielsen coincidence number $${N(\phi : \psi)}$$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to $${(\phi, \psi)}$$ . We calculate $${N(\phi : \psi)}$$ for all (m, n)-valued pairs of maps of the circle and show that $${N(\phi : \psi)}$$ is a sharp lower bound in that setting. Specifically, if $${\phi}$$ is of degree a and $${\psi}$$ of degree b, then $${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$$ , where $${\langle m, n \rangle}$$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.