Extending homeomorphisms from punctured surfaces to handlebodies

Let H g be a genus g handlebody and MCG 2 n ( T g ) be the group of the isotopy classes of orientation preserving homeomorphisms of T g = ∂ H g , fixing a given set of 2n points. In this paper we find a finite set of generators for E 2 n g , the subgroup of MCG 2 n ( T g ) consisting of the isotopy classes of homeomorphisms of T g admitting an extension to the handlebody and keeping fixed the union of n disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by the authors for n = 1 . The subgroup E 2 n g turns out to be important for the study of knots and links in closed 3-manifolds via ( g , n ) -decompositions. In fact, the links represented by the isotopy classes belonging to the same left cosets of E 2 n g in MCG 2 n ( T g ) are equivalent.