  Several new combinatorial descriptions of closed 4-manifolds have recently
been introduced in the study of smooth maps from 4-manifolds to surfaces. These
descriptions consist of simple closed curves in a closed, orientable surface
and these curves appear as so called vanishing sets of corresponding maps. In
the present paper we focus on homotopies canceling pairs of cusps so called
cusp merges. We first discuss the classification problem of such homotopies,
showing that there is a one-to-one correspondence between the set of homotopy
classes of cusp merges canceling a given pair of cusps and the set of homotopy
classes of suitably decorated curves between the cusps. Using our
classification, we further give a complete description of the behavior of
vanishing sets under cusp merges in terms of mapping class groups of surfaces.
As an application, we discuss the uniqueness of surface diagrams, which are
combinatorial descriptions of 4-manifolds due to Williams, and give new
examples of surface diagrams related with Lefschetz fibrations and Heegaard
diagrams.
