On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds

Given two topological spaces, is it possible to determine whether they are homeomorphie ? This is the homeomorphism problem and most work in topology is directed toward some aspect of the homeomorphism problem. A plan for solving the homeomorphism problem for "most" 3-manifolds has been developed by Wolfgang Haken. However, a certain very special step in this plan has eluded proof. The problem of providing a proof for this special case amounts to the problem of classifying homeomorphisms of compact, orientable 2-manifolds. In this paper a method for classifying homeomorphisms of compact, orientable 2-manifolds will be given, and hence it will be possible to classify all compact, orientable, irreducible, boundary irreducible, sufficiently large 3-manifolds. Hence "most" 3-manifolds of interest can be classified, including all knot and linl~ spaces. Haken developed the theory in his series of papers: [1]-[5]. In [11], Schubert has explained the essential points of Haken's theory of normal surfaces. Waldhausen [12] has written a summary of the classification procedure, using the recent results of Johannson [6], [7].