Mathematical Sciences: Essential Laminations in 3-Manifolds

基本层理是最近定义的不可压缩曲面和紧化3流形的无Reeb分量的协维1叶理的推广。这两个更经典的对象在从同伦理论数据中获得关于3流形的几何/拓扑信息方面都被证明是非常有用的，并且基本层压已经开始显示自己是它的两个“父母”的有价值的继承者。研究者打算用本质层积来研究3-流形的基群在多大程度上决定流形达到同胚，并研究流形是否允许双曲结构的问题。他进一步打算利用基本层合正规形式的存在性来研究3-流形（主要是双曲3-流形）是否包含基本层合的问题。令人惊讶的是，尽管我们生活在一个三维空间，一个所谓的三维流形中，因此我们对这些几何物体有一种天生的直觉，但最终这并没有把我们带得像我们所期望的那样远，因为那些在高维流形中通过代数计算已经解决的问题，在三维情况下仍然令人困惑。其中最著名的是庞加莱在世纪之交提出的关于三维球体的著名猜想，正是在这个猜想中，最初的三维情况是唯一仍然开放的。研究者正在研究关于三维流形的各种问题，其中一些关于距离的概念有点奇怪，即所谓的双曲度量，但这些问题一次又一次地被证明与具有更熟悉的距离概念的流形的情况有明确的关联。

Essential laminations are a recently defined generalization of both the incompressible surface and the codimension-one foliation without Reeb components of a compact 3-manifold. Both of these more classical objects have proved remarkably useful in obtaining geometric/topological information about 3-manifolds from homotopy-theoretic data, and the essential lamination has already begun to show itself to be a worthy successor to both of its `parents.' The investigator intends to use essential laminations to study the extent to which the fundamental group of a 3-manifold determines the manifold up to homeomorphism, and also to study the question of whether or not the manifold admits a hyperbolic structure. He further intends to use the existence of normal forms for essential laminations to study the question of whether or not a 3-manifold (principally a hyperbolic 3-manifold) contains an essential lamination. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigator is pursuing a variety of questions about 3-dimensional manifolds, some with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.