Extending homeomorphisms from punctured surfaces to handlebodies
Abstract
Let $\textup{H}_g$ be a genus
g
handlebody and $\textup{MCG}_{2n}(\textup{T}_g)$ be the group of the isotopy classes of orientation preserving homeomorphisms of $\textup{T}_g=\partial\textup{H}_g$, fixing a given set of
2n
points. In this paper we find a finite set of generators for
E
g
2n
, the subgroup of $\textup{MCG}_{2n}(\textup{T}_g)$ consisting of the isotopy classes of homeomorphisms of $\textup{T}_g$ 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
g
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
g
2n
in $\textup{MCG}_{2n}(\textup{T}_g)$ are equivalent.