The index of coincidence Nielsen classes of maps between surfaces

For a given pair of closed orientable surfaces S-h, S-g and given integers d(1), d(2), one would like to find bounds for the index of the Nielsen coincidence classes among all possible pairs of maps (f(1), f(2)): S-h --> S-g where \deg(f(1))\ = d(1) and \deg(f(2))\ = d(2). We show that these bounds are infinite when h > g = 1, or when h greater than or equal to g > 1 and both d(i) < (h - 1)/(g - 1). We calculate these bounds when h = g and d2 = 1. We also consider the similar question for the root case, which is simpler, and we solve it completely. Few results are given when d(i) = (h - 1)/(g - 1) for either i = 1 or i = 2. (C) 2001 Elsevier Science B.V. All rights reserved.