4-Manifold topology and related topics

4-流形拓扑及相关主题

• 批准号: 1005304
• 负责人: Robert Gompf
• 金额: $ 16.74万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-15 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005304&HistoricalAwards=false
• 关键词: Manifold; topology; related; topics

A major thrust of the project concerns Cappell-Shaneson homotopy 4-spheres. These have long been considered the most likely counterexamples to the smooth 4-dimensional Poincare Conjecture. Recent work by Akbulut, and a simpler, more general approach by the PI, have shown that infinitely many of these are standard. The PI plans to extend his method further, trying to show that all CS-spheres are standard. He will also study other homotopy 4-spheres containing fibered 2-knots, accumulating evidence suggesting that the conjecture is true. The project includes various other avenues of research related to the PI's expertise. He will search for compact domains of holomorphy in complex surfaces, using one of his recent theorems. It now suffices to locate smoothly embedded compact 4-manifolds admitting suitable handle decompositions. He hopes to find (or rule out) examples such as pseudoconvex embeddings of preassigned homology spheres, and a compact domain of holomorphy in C^2 homotopy equivalent to the 2-sphere (violating a conjecture of Forstneric). A related theorem of the PI allows one to construct topological embeddings of 3-manifolds with a generalized pseudoconvexity property; he will investigate these in more detail. Another project is to study what manifolds of dimension 4 and higher can be realized as orbit spaces of vector fields in Euclidean space. Preliminary research shows that such manifolds must be simply connected, but many 4-manifolds with nontrivial 2-homology can arise.Various other investigations, concerning compact 4-manifolds, symplectic structures and Lefschetz pencils will also be pursued.The Poincare Conjecture for 3-manifolds has now been proved by Perelman, a full century after it was first proposed. Its generalization to higher dimensions was proved in the 1960's except in dimension 4. That last version was proved around 1981 by Freedman. However, the original conjecture was made for topological manifolds, so that one is allowed to crinkle the manifold in complicated ways. When manifolds are used in practice, in geometry, analysis, physics, economics and so on, one normally wants to be able to apply calculus, so one must disallow crinkling and work entirely with smooth manifolds. The smooth analog of the Poincare Conjecture has been understood in dimensions 4 since the 1960's, and is equivalent to the topological version (hence solved) in dimensions 4. However, the smooth 4-dimensional Poincare Conjecture is still mysterious, and is the last fundamental open question remaining from the initial heyday of manifold topology a half-century ago. There have been many potential counterexamples constructed, homotopy 4-spheres that might not be the standard 4-sphere, but none has been shown to actually be nonstandard. It has also been quite difficult to show that any of these examples are standard, but the PI has been in the forefront of research in this direction. His new methods have dispensed with a large family of potential counterexamples that were constructed in the 1970's. He intends to further investigate this problem, adding evidence that the conjecture may be true after all, in spite of the prevailing belief to the contrary in recent decades. He will also study other problems involving 4-manifolds and other classical mathematical objects.

该项目的一个主要目的是关于Cappell-Shaneson同伦4-球面。长期以来，这些一直被认为是对光滑的四维庞加莱猜想最有可能的反例。阿克布卢特最近的工作，以及PI的一种更简单、更通用的方法，已经表明其中无限多是标准的。PI计划进一步扩展他的方法，试图证明所有CS-球面都是标准的。他还将研究其他包含纤维2结的同伦4球体，积累证据表明这个猜想是正确的。该项目包括与PI的专业知识相关的各种其他研究途径。他将使用他最近的一个定理在复杂曲面上寻找全纯的紧域。它现在足以定位平稳嵌入的紧凑型4-歧管，允许适当的手柄分解。他希望找到(或排除)这样的例子，如预先指定的同调球面的伪凸嵌入，以及C^2同伦中等价于2-球面的全纯紧域(违反了Forstneric的一个猜想)。PI的一个相关定理允许人们构造具有广义伪凸性的3-流形的拓扑嵌入；他将更详细地研究这些。另一个项目是研究欧氏空间中哪些4维及更高维的流形可以实现为向量场的轨道空间。初步研究表明，这样的流形一定是单连通的，但可能会出现许多具有非平凡2-同调的4-流形。关于紧致4-流形、辛结构和Lefschetz铅笔的各种其他研究也将继续进行。3-流形的Poincare猜想在首次提出整整一个世纪后，现在已经被Perelman证明了。推广到高维的结果在20世纪60年代的S被证明，但在4维除外。最后一个版本是在1981年左右由Freedman证明的。然而，最初的猜想是针对拓扑流形的，因此允许人们以复杂的方式将流形皱缩。当流形在实践中使用时，在几何、分析、物理、经济学等领域，人们通常希望能够应用微积分，因此必须禁止皱缩，完全使用光滑的流形。光滑的庞加莱猜想自1960年S以来一直被理解到4维空间，并且等价于4维的拓扑版本(因此被解决)。然而，光滑的4维庞加莱猜想仍然是神秘的，是半个世纪前流形拓扑学最初全盛时期留下的最后一个基本悬而未决的问题。已经构建了许多潜在的反例，同伦的4球可能不是标准的4球，但没有一个被证明实际上是非标准的。也很难证明这些例子中的任何一个都是标准的，但PI一直处于这一方向研究的前沿。他的新方法去掉了上世纪70年代构建的一大类潜在的反例。他打算进一步研究这个问题，补充证据表明，尽管近几十年来普遍存在相反的看法，但这个猜想终究可能是真的。他还将研究涉及4-流形和其他经典数学对象的其他问题。