Three-dimensional manifolds and their Heegaard diagrams
James Singer · Transactions of the American Mathematical Society · 1933
One of the outstanding problems in topology today is the classification of n-dimensional manifolds, n >3. Poincare, the founder of modern analysis situs, devoted several papers to it and allied problems. t HeegaardT, in a paper concerned primarily with another aspect of the subject, found it convenient to construct a pseudo-normal form for a 3-dimensional manifold, a form which we now call the Heegaard diagram. Dehn? and Veblenll gave modifications of his construction. The Heegaard diagram of a 3-dimensional manifold consists of a closed 2-dimensional manifold upon which are drawn a certain number of nonintersecting simple closed curves. Any diagram is an adequate representation of a 3-dimensional manifold in the sense that it completely determines such a manifold, but, unfortunately, a 3-dimensional manifold gives rise to an infinity of diagrams. The problem of classifying manifolds is thus transferred to the problem of classifying diagrams. Heegaard, in the paper cited above, studied (although not completely) the modifications that can be made on the curves and surface of a diagram which transformed it into another diagram but yet did not change the manifold which it represented. In this paper we extend Heegaard's results and study more completely the relationships between manifolds and their diagrams. We begin then (Part I) by introducing the notions of a canonical region, canonical surface and canonical curve of a manifold. The Heegaard diagram is then constructed from a canonical surface and curves. Then, before proceeding to a discussion of manifolds and their diagrams, we show (Part II) how to read off the usual invariants of a manifold from any one of its representative diagrams.