On the Mapping class group of a genus 2 handlebody

Abstract

A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.