Mathematical Sciences: Low-Dimensional Topology and Gauge Theory

数学科学：低维拓扑和规范论

• 批准号: 9704204
• 负责人: Nikolai Saveliev
• 金额: $ 4.49万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 1998-09-24
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704204&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Low; Dimensional; Topology

9704204 Saveliev The research is concerned with a topic which is central to the theory of smooth 4-dimensional manifolds, namely the homology cobordism group of integral homology 3-spheres. The long-term goal is to make progress in solving the long-standing problem of R. Kirby about the existence of elements of order two in this group carrying the Rohlin invariant. The investigator concentrates his attention at the elements represented by the links of algebraic singularities, in particular, Seifert fibered homology spheres. His approach is based on application of the methods of modern gauge theory to investigation of the invariant introduced in the 70's by W. Neumann and L. Siebenmann for the so-called graph homology 3-spheres. This application is threefold. First, he uses some special resolutions of singularities to construct a 4-cobordism of an algebraic link with the prescribed intersection form, and he then applies the gauge-theoretical results which prohibit certain integral bilinear forms as intersection forms of such a resolution if the link bounds a homology ball. The result of this investigation is further applied to show that Rohlin invariant one algebraic links cannot have order two in the homology cobordism group. Second, the investigator uses the instanton Floer homology to introduce a new invariant for arbitrary homology 3-spheres, and he proves that it agrees with the invariant of Neumann and Siebenmann when the latter is defined (graph manifolds). This result also gives new insights into Floer homology, in particular, answers positively M. Atiyah's question whether there is a Milnor fiber description of Floer homology of certain algebraic links, and whether it is related to the complex conjugation action on the homology of the Milnor fiber. Finally, the investigator applies his new invariant to investigate Kirby's problem for general homology spheres. The first step in this direction is to prove that his invariant should vanish on all hom ology 3-spheres that bound a homology ball with just one 1-handle (and a 0- and 2-handle). His technique is to use the Floer exact triangle to keep track of the Floer homology along the cobordism. One of the central objects of investigation in both mathematics and theoretical physics is a smooth n-dimensional manifold. Although this object looks locally like an n-dimensional Euclidean space, its global structure may still be very rich and complicated. Most major results about 2-dimensional manifolds were obtained in the 19th century. Manifolds in dimensions greater than or equal to 5 were successfully classified in the 1960's. Though some major questions about 3-manifolds remain unanswered, it is manifolds of dimension 4, the dimension of relativity theory, that occupy the most special place in the manifold hierarchy. On the one hand, they are not "big enough" to apply to them the arguments that proved to be so useful in higher dimensions. On the other hand, their dimension is too big to apply more intuitive methods that work effectively in lower dimensions. Progress has been slow here for a few decades until recent developments that involved the application of ideas from the physics of gauge theories. The main results are due to S. Donaldson, who initiated the whole program in the early 1980's, and most recently to N. Seiberg and E. Witten. The gauge-theoretical approach proved to be very fruitful and led to the solution of many hard problems. The investigator is applying these modern methods to make progress on yet another long-standing problem in topology, the structure of the homology cobordism group of homology 3-spheres, which would provide new insight into manifold structure. ***

9704204 Saveliev 该研究涉及光滑 4 维流形理论的核心主题，即积分同调 3 球体的同调配边群。 长期目标是在解决 R. Kirby 长期存在的问题上取得进展，该问题涉及带有罗林不变量的群中二阶元素的存在。 研究人员将注意力集中在代数奇点链接所代表的元素上，特别是 Seifert 纤维同调球。 他的方法基于应用现代规范理论方法来研究 W. Neumann 和 L. Siebenmann 在 70 年代针对所谓的图同调 3-球体引入的不变量。 该应用程序有三重。 首先，他使用奇点的一些特殊分辨率来构造具有规定交集形式的代数链接的 4-共边，然后他应用规范理论结果，该结果禁止某些积分双线性形式作为此类分辨率的交集形式，如果链接限制同源球。 进一步应用这一研究结果表明Rohlin不变一代数链在同调共边群中不能有二阶。 其次，研究者使用瞬时Floer同调为任意同调3-球体引入了一个新的不变量，并且证明了当后者被定义时（图流形），它与Neumann和Siebenmann的不变量一致。 这一结果也对Floer同调给出了新的认识，特别是正面回答了M. Atiyah的问题：某些代数环节的Floer同调是否存在Milnor纤维描述，是否与Milnor同调上的复杂共轭作用有关纤维。 最后，研究人员应用他的新不变量来研究一般同调球的柯比问题。 这个方向的第一步是证明他的不变量应该在所有仅用一个 1 手柄（以及 0 和 2 手柄）约束同源球的同源 3 球上消失。 他的技术是使用弗洛尔精确三角形来跟踪沿共边的弗洛尔同源性。 数学和理论物理研究的中心对象之一是光滑的 n 维流形。 尽管这个对象局部看起来像一个n维欧几里得空间，但它的全局结构可能仍然非常丰富和复杂。 关于二维流形的大多数主要成果是在 19 世纪获得的。 维数大于或等于 5 的流形在 1960 年代被成功分类。 尽管有关 3 流形的一些主要问题仍未得到解答，但相对论维度 4 的流形在流形层次结构中占据着最特殊的位置。 一方面，它们还“不够大”，无法应用在更高维度中被证明如此有用的论据。 另一方面，它们的维度太大，无法应用在较低维度下有效工作的更直观的方法。 几十年来，这方面的进展一直很缓慢，直到最近的发展涉及规范理论物理学思想的应用。 主要成果归功于 S. Donaldson，他在 1980 年代初发起了整个计划，最近的成果归功于 N. Seiberg 和 E. Witten。 规范理论方法被证明是非常富有成效的，并解决了许多难题。 研究人员正在应用这些现代方法来解决拓扑学中另一个长期存在的问题，即同调 3-球的同调配边群的结构，这将为流形结构提供新的见解。 ***