Extending homeomorphisms from punctured surfaces to handlebodies

2007/02/20 by Alessia Cattabriga, Cattabriga, Alessia, Michele Mulazzani +1
Mathematics · #20F38 #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0702570

openalex publication_date 2007/02/20 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let \textupH_g be a genus g handlebody and \textupMCG_2n(\textupT_g) be the group of the isotopy classes of orientation preserving homeomorphisms of \textupT_g=∂\textupH_g, fixing a given set of 2n points. In this paper we find a finite set of generators for E_2n^g, the subgroup of \textupMCG_2n(\textupT_g) consisting of the isotopy classes of homeomorphisms of \textupT_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_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 E_2n^g in \textupMCG_2n(\textupT_g) are equivalent.

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