Low-Dimensional Manifolds

9704490 Luo Luo studies topology and geometry of surfaces and Heegaard splittings of 3-manifolds. One of the fundamental tools in surface theory is the space of isotopy classes of essential simple loops on surfaces introduced by Dehn and Thurston. Dehn called the space the arithmetic field of the surface. The space was used by Thurston in his work on the compactification of the Teichmueller space. It has been known by many mathematicians that the space of isotopy classes has an intrinsic modular structure. Luo has used the modular structure to reconstruct the Teichmueller space and the space of measured laminations. This has enabled him to derive that the space of measured laminations is semi-real algebraic. He is currently working on the reconstruction of the mapping class group of the surface. He is also trying to apply the result on the space of measured laminations to study the Heegaard diagrams of irreducible non-Haken 3-manifolds. Many problems in topology, geometry and mathematical physics involve the consideration of all simple loops on a surface. An essential simple loop is a curve without self-crossing that cannot be shrunk to a point on the surface. The problem is to understand all such loops put together. The basic relation between loops is their intersection number. Luo is developing an algebraic calculation to understand the intersection numbers on a space. This may eventually lead to a better understanding of 3-dimensional spaces through the use of Heegaard surface theory. ***