A Complex of Incompressible Surfaces for handlebodies and the Mapping   Class Group
  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. 
Charalampos Charitos (1), Ioannis Papadoperakis (1) and Georgios   Tsapogas (2) ((1) Agricultural University of Athens and (...
math.GT