Complete systems of surfaces in 3-manifolds

Abstract

A complete system (CS) [Jscr]={J₁, ..., J_n} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr] is a CS of surfaces for M, then any CS equivalent to [Jscr] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds: