Mathematical Sciences: "Knot Theory and 3-Manifolds"

该研究是在结理论和3- 维流形 汤普森将要解决的问题之一 是P性质猜想 在三维球面中的一个纽结被称为 如果在结上没有非平凡的手术产生一个 同伦三维球面 长期以来，人们一直认为，所有非- 3-球面中的平凡纽结具有P. Gordon和Luecke性质 最近的研究表明， 纽结本身不能产生三维球体。 仍有 可以对3球中的结进行手术的可能性 并得到庞加莱猜想的反例。 的 对这一问题的建议方法涉及到一个概括， 工作的Gabai缝合流形，以及考虑 一个等价的问题，一个人是否可以做手术的结在一个 伪3-球面并获得S3。 此外，最近与 Scharlemann提出了几个有趣的问题， 3-流形中的嵌入有限图 汤普森会尝试 将他们的工作推广到刻画无结平面图， S3推广到3-球面中的任意有限图。 这些都是关于三维流形的基本问题。 的 它们是开放的这一事实表明我们仍然不知道多少 关于三维流形。 尽管事实上 我们生活在其中，所以这样的拓扑问题甚至可能 具有宇宙学意义。

The research is in the areas of knot theory and 3- dimensional manifolds. One of the problems Thompson will work on is the property-P conjecture. A knot in the 3-sphere is said to have property P if no non-trivial surgery on the knot yields a homotopy 3-sphere. It has long been conjectured that all non- trivial knots in the 3-sphere have property P. Gordon and Luecke have recently shown that non-trivial surgery on a non-trivial knot cannot yield the 3-sphere itself. There remains the possibility that one could do surgery on a knot in the 3-sphere and obtain a counter-example to the Poincare conjecture. The suggested approach to this problem involves a generalization of work of Gabai on sutured manifolds, as well as considering the equivalent problem of whether one can do surgery on a knot in a fake 3-sphere and obtain S3. In addition, recent work with Scharlemann has suggested several interesting questions about imbedded finite graphs in 3-manifolds. Thompson will try to generalize their work characterizing unknotted planar graphs in S3 to arbitrary finite graphs in the 3-sphere. These are fundamental questions about three-manifolds. The fact that they are open points up how much we still do not know about three-dimensional manifolds. This is so despite the fact that we live in one, and so such topological questions might even have cosmological significance.