Maps between surfaces and minimal coincidence sets for homotopies

In this paper we consider the 1-parameter coincidence problem of finding homotopies of pairs of maps (ft,gt) such that the number of coincidence points is independent of t. A number of results for the root problem (gt constant) for maps between surfaces are given. Also, for the fixed point problem (gt the identity) on the 2-sphere we obtain a negative result for every degree different from negative one. That is, the existence of pairs f0 and f1 each having one fixed point, but for some t, ft has additional fixed points.