Extending Homeomorphisms from Punctured Surfaces to Handlebodies

Let Hg be a genus g handlebody and MCG2n(Tg) be the group of the isotopy classes of orientation preserving homeomorphisms of Tg = ∂Hg, fixing a given set of 2n points. In this paper we find a finite set of generators for E 2n, the subgroup of MCG2n(Tg) consisting of the isotopy classes of homeomorphisms of Tg 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 2n 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 2n in MCG2n(Tg) are equivalent. Mathematics Subject Classification 2000: Primary 20F38; Secondary 57M25.