Heegaard Splittings, Knots and 3-Manifolds

Heegaard 分裂、结和 3 流形

• 批准号: 1207765
• 负责人: Abigail Thompson
• 金额: $ 21.86万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207765&HistoricalAwards=false
• 关键词: Heegaard; Splittings; Knots; Manifolds

An overarching goal in the study of 3-manifolds, post-Perelman, is to put them in some satisfactory order. One model for an order can be found in 2-dimensions; surfaces, orientable or not, with or without boundary, come in a neat package, which is well-understood. Thanks to Thurston and Perelman one now knows inexpressibly more than before about the geometry of all 3-manifolds, yet one still seeks an elegant framework in which to place them all. This quest leads to some long-standing problems, as well as introducing some new ones. This proposal describes problems in this general area which can be considered long-standing, although from a new point of view. For example, the PI will build on the fundamental work on the stabilization problem for Heegaard splittings which was completed during the previous grant, working with J. Hass and W. Thurston, and proposes to complete the second half of this conjecture. The PI introduces a new knot invariant, related to earlier joint research with Scharlemann, and suggests methods to answer some very basic questions regarding it. In addition the PI includes a problem closely related to the long-standing slice-ribbon conjecture. The PI also discuss her extensive outreach activities, which are intimately entwined with her research.A 3-dimensional manifold is an object that looks, to a local observer, like 3-dimensional Euclidean space. That is, it looks like the room in which you are likely to be sitting. But just as several thousand years ago mankind knew only the local, not the global, shape of the earth, at present we know only about the local, not the global, shape of the 3-dimensional universe in which we live. So 3-dimensional spaces are in many respects important objects of study. Tremendous progress in this field has been made in recent years, and there is now a hope of understanding the full sweep of 3-dimensional spaces. The problems in this proposal address some of the longstanding structural questions in the field, with the aim of uncovering at least part of what is sure to be, when it is ultimately discovered, the elegant framework which underlies it.

在后佩雷尔曼时代，三维流形研究的首要目标是将它们排列成令人满意的顺序。一个订单的一个模型可以在二维中找到;表面，可定向或不可定向，有或没有边界，在一个整洁的包装中，这是很好理解的。由于瑟斯顿和佩雷尔曼现在知道的几何形状的所有3-流形难以形容的比以前，但人们仍然寻求一个优雅的框架，将它们全部。这种追求导致了一些长期存在的问题，也带来了一些新的问题。本提案说明了这一一般领域的问题，这些问题可以被认为是长期存在的，尽管是从一个新的角度提出的。例如，PI将建立在Heegaard分裂的稳定问题的基础工作上，这是在上一次资助期间完成的，与J. Hass和W。Thurston，并建议完成这个猜想的后半部分。PI介绍了一个新的结不变量，与早期的联合研究与Scharlemann，并建议的方法来回答一些非常基本的问题，此外，PI包括一个问题密切相关的长期切片带猜想。PI还讨论了她广泛的外展活动，这些活动与她的研究密切相关。三维流形是一个物体，对于当地观察者来说，就像三维欧几里得空间。也就是说，它看起来像你可能坐的房间。但是，正如几千年前人类只知道地球的局部形状，而不是全球形状一样，现在我们只知道我们所生活的三维宇宙的局部形状，而不是全球形状。所以三维空间在很多方面都是重要的研究对象。近年来，这一领域取得了巨大的进展，现在有希望了解三维空间的完整扫描。本提案中的问题解决了该领域一些长期存在的结构性问题，目的是至少揭示一部分最终被发现时肯定会成为其基础的优雅框架。