Filling of closed surfaces

Abstract

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.