Extending homeomorphisms from punctured surfaces to handlebodies

Details Extending homeomorphisms from punctured surfaces to handlebodies Extending homeomorphisms from punctured surfaces to handlebodies

2007-02-20

arxiv.org/abs/math/0702570v2

Mathematics

Geometric Topology

We correct the statements of Theorem 9 and 10, by adding missing generators, and improve the statement of Theorem 10, by remov

Scientific paper

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 $\mathcal{E}_{2n}^g$, 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 $\mathcal{E}_{2n}^g$ 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 $\mathcal{E}_{2n}^g$ in $\textup{MCG}_{2n}(\textup{T}_g)$ are equivalent.

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