Spaces of Incompressible Surfaces

The homotopy type of the space of PL homeomorphisms of a Haken 3 manifold was computed in [H1], and with the subsequent proof of the Smale conjecture in [H2], the computation carried over to diffeomorphisms as well. These results were also obtained independently by Ivanov [I1,I2]. The main step in the calculation in [H1], though not explicitly stated in these terms, was to show that the space of embeddings of an incompressible surface in a Haken 3 manifold has components which are contractible, except in a few special situations where the components have a very simple noncontractible homotopy type. The purpose of the present note is to give precise statements of these embedding results along with simplified proofs using ideas from [H3]. We also rederive the calculation of the homotopy type of the diffeomorphism group of a Haken 3 manifold.