Low-dimensional topology and the complex of curves

三维拓扑学是最容易理解的数学领域，这是因为三维空间是日常经验的领域。庞加莱在19世纪后期发表的一系列鼓舞人心的文章中确立了低维拓扑学的大部分工作是组合性的。1979年，瑟斯顿通过揭示与许多其他更几何的数学领域的深刻联系，彻底改变了这个领域。低维拓扑中的几何主题已经扩展到包括粗几何。这一主题的一个惊人的成功是工作的Masur和Minsky [1999年，2000年];他们介绍了使用粗糙的几何理解复杂的曲线的想法。Harvey定义的曲线复形是Teichmuller空间的一个组合类似物，本文提出的新奇是用Masur和Minsky的方法来研究和解决低维拓扑中的组合问题，其中有些是Poincare首次遇到的。贯穿该建议的主题包括天体和Heegaard图，三球面的投影问题，映射类群的结构，以及曲面自同构的配边群。

Topology in dimension three is the most accessible field ofmathematics; this is because three-dimensional space is the realm ofeveryday experience. Established by Poincare in a series of inspiringarticles in the late 1800's much of the work in low-dimensionaltopology was combinatorial in nature. In 1979 Thurston revolutionizedthe field by revealing deep connections to many other, more geometric,areas of mathematics.The geometric theme in low-dimensional topology has expanded to include coarse geometry. One striking success of this theme was the work of Masur and Minsky [1999, 2000]; they introduced the idea of using coarse geometry to understand the complex of curves. The complex of curves, defined by Harvey, is a combinatorial analogue of Teichmuller space.The novelty proposed here is to use the methods of Masur and Minsky tostudy and solve combinatorial problems, some first encountered byPoincare, in low-dimensional topology. The themes running through theproposal include handlebodies and Heegaard diagrams, the recognitionproblem for the three-sphere, the structure of the mapping classgroup, and the cobordism group of automorphisms of surfaces.