4-Manifold topology and related topics

该项目的一个主要推力涉及Cappell-Shaneson同伦四球。这些一直被认为是平滑四维庞加莱猜想最有可能的反例。Akbulut最近的工作，以及PI的一种更简单、更一般的方法，已经表明其中无限多个是标准的。PI计划进一步扩展他的方法，试图证明所有的cs球都是标准的。他还将研究其他含有2节纤维的同伦4球，以积累证据来证明这个猜想是正确的。该项目包括与PI专业知识相关的各种其他研究途径。他将利用他最近的一个定理，在复杂曲面上寻找全纯的紧致区域。它现在足以定位顺利嵌入紧凑型4流形承认合适的处理分解。他希望找到（或排除）一些例子，如预先指定的同伦球的伪凸嵌入，以及C^2同伦中等价于2球的全纯紧致域（违反Forstneric的一个猜想）。PI的一个相关定理允许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问题。另一个课题是研究在欧氏空间中，四维及以上的流形可以实现为向量场的轨道空间。初步研究表明，这种流形必须是单连通的，但也可以出现许多具有非平凡2同调的4-流形。各种其他的研究，关于紧凑的4流形，辛结构和Lefschetz铅笔也将继续进行。3流形的庞加莱猜想在被提出整整一个世纪之后，现在已经被佩雷尔曼证明了。它在高维上的推广在60年代得到了证明，除了在四维上。最后一个版本是在1981年由弗里德曼证明的。然而，最初的猜想是针对拓扑流形提出的，因此人们可以以复杂的方式使流形起皱。当流形在几何学、分析学、物理学、经济学等领域被实际应用时，人们通常希望能够应用微积分，所以必须禁止起皱，完全使用光滑的流形。自20世纪60年代以来，庞加莱猜想的光滑模拟已经在4维中得到了理解，并且等同于4维中的拓扑版本（因此得到了解决）。然而，光滑的四维庞加莱猜想仍然是神秘的，并且是半个世纪前流形拓扑学鼎盛时期遗留下来的最后一个基本开放问题。有许多潜在的反例被构造出来，同伦4球可能不是标准的4球，但没有一个被证明是非标准的。要证明这些例子中的任何一个都是标准的也相当困难，但PI一直处于这个方向的研究前沿。他的新方法省去了20世纪70年代构建的一大批潜在的反例。他打算进一步研究这个问题，增加证据来证明这个猜想可能毕竟是正确的，尽管近几十年来普遍认为与之相反。他还将研究涉及4流形和其他经典数学对象的其他问题。

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.