Nielsen theory of roots of maps of pairs

A relative root Nielsen number Nrel(ƒ; c) is introduced which is a homotopy invariant lower bound for the number of roots at c for a map of pairs of spaces ƒ : (X, A) → (Y, B) and c ϵ Y. Conditions are given which ensure that Nrel(ƒ; c) is a sharp lower bound. The standard method for the computation of the root Nielsen number N(ƒ; c) from the homomorphism of the fundamental group induced by ƒ is extended to obtain formulae for Nrel(ƒ; c). Many of the results depend on the location of c.