A Complex of Incompressible Surfaces for handlebodies and the Mapping Class Group
Charalampos Charitos (1), Ioannis Papadoperakis (1), Georgios, Tsapogas (2) ((1) Agricultural University of Athens, (2) University of the, Aegean)
TL;DR
This paper constructs a simplicial complex of incompressible surfaces in a handlebody, explores its properties, and shows its automorphism group is isomorphic to the handlebody's mapping class group, extending classical surface theory.
Contribution
It introduces a new complex of incompressible surfaces for handlebodies and establishes its automorphism group as the handlebody's mapping class group.
Findings
The complex contains the curve complex of the boundary surface as a subcomplex.
Automorphisms of the complex correspond to the handlebody's mapping class group.
Properties of the complex are rigorously established.
Abstract
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the boundary surface of H. As in the classical theory, the group of automorphisms of this complex is identified with the mapping class group of the handlebody.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis