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

我们生活在一个四维的宇宙中，把时间当作一个维度，然而令人惊讶的是，我们对四维空间的几何学知之甚少。拓扑学研究的是关于空间的真正基本的几何问题，这些问题即使通过平滑地弯曲和变形空间结构也不会改变。举一个三维的例子，一个带有方结的弦环与一个带有奶奶结的弦环在拓扑上是不同的，在这个意义上，弦或它周围的空间的任何光滑变形都不能将一个环变成另一个环;研究这种打结是典型的三维拓扑问题。在四维空间中，打结也会以多种不同的方式发生，它也是光滑四维拓扑结构的核心，因此也是理解我们生活的空间的形状和解剖结构的核心。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，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内的运动。这项研究也与外展部分密切相关，主要研究者将与设计专业人士和当地社区成员合作，为拓扑学中的重要思想创造令人兴奋的可视化效果，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4维Poincare和Schoenflies拓扑。（庞加莱：每一个同伦4-球面都是标准4-球面的同胚。Schoenflies：在4-空间中的每一个光滑嵌入的3-球都有一个光滑嵌入的4-球。与之相关的一个基本问题是四维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说什么;基本的问题是，似乎有“没有坚持”时，与球，甚至同伦球;低维拓扑需要新的想法。在合作首先与柯比，后来与一个更大的小组，PI，在过程中的几个早期的NSF奖，制定了研究的莫尔斯2职能，推广莫尔斯理论，然后制定了理论的三分之四流形作为一个惊人的四维模拟理论的Heegaard分裂。这似乎正是需要注入新的生命到4-流形拓扑，并允许广泛的交叉施肥之间的世界3流形和4-流形。特别是在过去的五年（2014-19）中，其他研究人员对三分法的兴趣激增，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合作的拟议项目建立在这项工作的基础上，使用了许多与三分法工作相同的工具。这里有两个很有启发性的例子：可以想象，理解4-球面上所有三等分的空间将使我们能够证明某些自同构不是恒等式的同位素，并且可能将4-球面与其他同伦4-球面区分开来。同样，通过三等分和开卷分解，我们可以更深入地理解接触拓扑与四维流形拓扑的关系，这将使我们能够看到四维空间中的每个嵌入的三维球面都是接触型的，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

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.