The minimal number of singularities of stable maps between surfaces

For a C ∞ stable map φ : M → S 2 of a closed and connected surface M into the sphere, let c ( φ ) , i ( φ ) and n ( φ ) denote the numbers of cusps, fold curves components and nodes respectively. In this paper, in a given homotopy class, we will determine the minimal pair ( i , c + n ) with respect to the lexicographic order.