Apparent contours of stable maps of surfaces with boundary into the plane

. Let M be a connected compact surface with boundary. A C ∞ map M → R 2 is admissible if it is non-singular on a neighborhood of the boundary. For a C ∞ stable map f : M → R 2 , denote by c ( f ) and n ( f ), i ( f ) the number of cusps and nodes, connected components of the set of singular points respectively. In this paper, we introduce the notion of admissibly homotopic among C ∞ maps M → R 2 , and we will determine the minimal number c + n for each admissibly homotopy class.