Satellites of an oriented surface link and their local moves

For an oriented surface link F in R 4 , we consider a satellite construction of a surface link, called a 2-dimensional braid over F, which is in the form of a covering over F. We introduce the notion of an m-chart on a surface diagram π ( F ) ⊂ R 3 of F, which is a finite graph on π ( F ) satisfying certain conditions and is an extended notion of an m-chart on a 2-disk presenting a surface braid. A 2-dimensional braid over F is presented by an m-chart on π ( F ) . It is known that two surface links are equivalent if and only if their surface diagrams are related by a finite sequence of ambient isotopies of R 3 and local moves called Roseman moves. We show that Roseman moves for surface diagrams with m-charts can be well-defined.