Smooth 4-Manifolds: 2-3, 5- and 6-Dimensional Perspectives

我们生活在一个四维的宇宙中，把时间视为一个维度，但令人惊讶的是，我们对四维空间的几何学知之甚少。拓扑学研究关于空间的真正的基本几何问题，这些问题即使在平滑地弯曲和变形空间结构的情况下也是不变的。作为一个三维的例子，一个系着正方形结的弦环与一个系着奶奶结的弦环在拓扑上是不同的，因为无论弦或它周围的空间有多少平滑的变形，都不能把一个环变成另一个环；研究这种结是典型的三维拓扑问题。结节也发生在四维空间中，以无数不同的方式，也是平滑的四维拓扑结构的核心，因此确实是理解我们一生所居住的那种空间的形状和解剖的核心。该奖项支持的研究将通过从多个不同维度观察四维拓扑来探索这些问题，范围从仔细研究四维空间及其子空间到二维空间的投影，到仔细研究四维空间在五维和六维空间中的运动。这项研究还与一个外展部分密切相关，在这个部分中，首席调查员将与设计专业人员和当地社区成员合作，创建令人兴奋的拓扑学中重要思想的可视化，以扩大公众对这些迷人主题的认识和兴趣。此外，该项目还为研究生提供了研究培训的机会。拓扑学中两个最大的遗留问题是光滑的四维Poincare猜想和勋蝇猜想。(Poincare：每个同伦的4-球面都与标准的4-球面微分同胚。苍蝇：每个平滑嵌入4空间边界的3球就是一个平滑嵌入的4球。)一个相关的基础性问题是4-球面的光滑映射类群是否平凡。现有的技术，如规范理论，在这些问题的背景下什么都没有说；基本的问题是，在处理球体甚至同伦球体时，似乎没有什么可以坚持的；低维拓扑需要新的想法。在最初与Kirby和后来与一个更大的小组的合作中，PI在早期的几个NSF奖项的过程中，发展了对Morse 2-函数的研究，推广了Morse理论，然后发展了4维流形的三分理论，作为Heegaard分裂理论的惊人的4维模拟。这似乎正是为4-流形拓扑注入新生命，并允许3-流形和4-流形的世界之间广泛的交叉受精所需要的。特别是在过去的五年里(2014-19年)，其他研究人员对三分割法的兴趣激增，参与这项工作的社区也显著扩大。PI在接触和辛拓扑方面也有很强的背景；与LICATA一起提出的项目建立在这项工作的基础上，使用了许多在三分点工作中使用的相同工具。这里有两个不太可能的激励例子：可以想象，理解4-球面上所有三分线的空间将允许我们证明某些自微分同态与恒等式不同，并且还可能将4-球面与其他同伦的4-球面区分开来。同样，通过更深入地了解与4-流形拓扑相关的接触拓扑学，可能通过三分割法和开卷分解，将允许我们将4-空间中的每个嵌入的3-球体视为接触型的，从而允许我们使用辛工具来解决舍恩费尔斯猜想。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

We live in a 4-dimensional universe, counting time as a dimension, and yet surprisingly we understand very little about the geometry of 4-dimensional spaces. Topology studies truly fundamental geometric issues about space that are unchanged even by smoothly bending and deforming the structure of space. As an example from three dimensions, a loop of string with a square knot tied into it is topologically distinct from a loop of string with a granny knot tied into it, in the sense that no amount of smooth deformation of the string or the space around it can turn one loop into the other; studying this kind of knottedness is the quintessential three-dimensional topological problem. Knottedness also happens in four dimensions, in numerous different ways, and is also at the core of smooth four-dimensional topology, and thus really at the core of understanding the shape and anatomy of exactly the kind of space that we spend our lives inhabiting. The research supported by this award will probe these problems by looking at four-dimensional topology from multiple different dimensions, ranging from a careful study of projections of four-dimensional spaces and their subspace onto two dimensional spaces to a careful study of motions of four-dimensional spaces inside five- and six-dimensional spaces. This research is also closely related to an outreach component, in which the principal investigator will work with design professionals and members of the local community to create exciting visualizations of important ideas in topology, to broaden the public's awareness of and interest in these fascinating topics. In addition the project provides research training opportunities for graduate students.Two of the biggest remaining problems in topology are the smooth 4-dimensional Poincare and Schoenflies conjectures. (Poincare: Every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. Schoenflies: Every smoothly embedded 3-sphere in 4-space bounds a smoothly embedded 4-ball.) A related and foundational problem is whether the smooth mapping class group of the 4-sphere is trivial. Existing techniques such as gauge theory say nothing in the context of these problems; the basic problem is that there seems to be "nothing to hold on to" when working with spheres or even homotopy spheres; low-dimensional topology needs new ideas. In collaboration first with Kirby and later with a larger group, the PI, over the course of several earlier NSF awards, developed the study of Morse 2-functions, generalizing Morse theory, and then developed the theory of trisections of 4-manifolds as a striking 4-dimensional analog to the theory of Heegaard splittings. This seems to be exactly what is needed to inject new life into 4-manifold topology and to allow for extensive cross-fertilization between the worlds of 3-manifolds and 4-manifolds. The past five years (2014-19), in particular, have seen an explosion of interest by other researchers in trisections, and a significant broadening of the community involved in this work. The PI also has a strong background in contact and symplectic topology; the proposed project with Licata builds on this work, using many of the same tools used in the work on trisections. Here are two long-shot motivational examples: It is conceivable that understanding the space of all trisections on the 4-sphere will allow us to show that certain self-diffeomorphisms are not isotopic to the identity, and also perhaps distinguish the 4-sphere from other homotopy 4-spheres. Likewise it is possible that a deeper understanding of contact topology in relation to 4-manifold topology, perhaps via trisections and open book decompositions, will allow us to see every embedded 3-sphere in 4-space as being of contact type, allowing us to use symplectic tools to settle the Schoenflies conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.