NIELSEN COINCIDENCE, FIXED POINT AND ROOT THEORIES OF n-VALUED MAPS

Let (φ, ψ) be an (m,n)-valued pair of maps φ, ψ : X ( Y , where φ is an m-valued map and ψ is n-valued, on connected finite polyhedra. A point x ∈ X is a coincidence point of φ and ψ if φ(x) ∩ ψ(x) 6= ∅. We define a Nielsen coincidence number N(φ : ψ) which is a lower bound for the number of coincidence points of all (m,n)-valued pairs of maps homotopic to (φ, ψ). We calculate N(φ : ψ) for all (m,n)-valued pairs of maps of the circle and show that N(φ : ψ) is a sharp lower bound in that setting. Specifically, if φ is of degree a and ψ of degree b, then N(φ : ψ) = |an− bm| < m,n > , where < m,n > is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for nvalued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown. Subject Classification 55M20, 54C60