Mathematical Sciences: Research In Algebraic Topology

数学科学：代数拓扑研究

• 批准号: 9303489
• 负责人: Edgar Brown
• 金额: $ 18万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303489&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Algebraic; Topology

Four investigators will study a variety of problems in algebraic topology. (1) Edgar H. Brown, Jr. will continue his applications of the machinery developed in "Continuous Cohomology and Real Homotopy Theory" to mod p homotopy theory, classifying spaces of foliations, and real homotopy type of Kaehler manifolds. (2) Kiyoshi Igusa will study the relationship between the higher Reidemeister torsion invariants (algebraic K-theory invariants associated to the diffeomorphism space of a smooth manifold) and the simpler cyclic homology groups associated to circle bundles. In particular, he has a plan to prove a complexified version of a theorem of Bokstedt, which if proved would give an explicit realization of these difficult invariants. The methods he will use are Morse theory, algebraic K-theory, and cyclic homology. (3) Jerome P. Levine will work on a direct topological description of the Atiyah-Patodi-Singer rho-invariant of an odd-dimensional manifold, and determining when it is a homotopy invariant. In particular, he has a conjectured formula for the jump along a path of representations. He is also investigating the possible use of the algebraic version of the rho-invariant (a multisignature) to determine the torsion-free part of the homology concordance surgery group (Gamma-group) of Cappell-Shaneson. He will continue to study the "algebraic closure" construction in group theory, particularly in its application to link concordance questions. Several geometric questions reduce to purely algebraic questions about the algebraic closure of the free group. (4) Daniel Ruberman will treat a number of problems in gauge theory related to the topology of low dimensional manifolds. One main aspect of the work is the further development of gauge theory on 4-manifolds with boundary, via the moduli space of finite energy anti-self-dual connections. Applications include obstructions to the representation of homology classes by surfaces of low genus as well as a new knot cobordism invariant. The project also includes a study of gauge-theoretic invariants of connections on 3-manifolds, and the relation of such invariants to the topology of the space of flat SU(2) connections. These invariants lead to potentially new obstructions to a group's being the fundamental group of a 3-manifold. These various projects concern the topology and geometry of manifolds, which are spaces locally resembling the familiar Euclidean spaces. Three and four-dimensional manifolds provide models for the physical world of space and time, while manifolds of higher dimension arise naturally as the parameter spaces for complicated systems involving many variables. The fundamental problems about such spaces include their classification in terms of numerical and algebraic invariants, the classification of their symmetries (or diffeomorphisms) and the determination of how one space may sit inside another (as a knotted curve sits inside of 3-dimensional space). Brown is perfecting some of the algebraic machinery required. Igusa's part of the project studies ways in which complicated families of symmetries may be understood in terms of algebraic invariants derived from corresponding families of matrices. Levine is investigating the interpretation in topology of an analytically defined invariant, as well its uses in the theory of knots. Ruberman will use the Yang-Mills equations of mathematical physics to understand the nature of four-dimensional manifolds and the two-dimensional surfaces which they contain.

四名研究人员将研究代数拓扑中的各种问题。 (1) Edgar H. Brown, Jr. 将继续将其在“连续上同调和实同伦理论”中开发的机制应用于 mod p 同伦理论、叶状结构空间分类和凯勒流形的实同伦类型。 (2) Kiyoshi Igusa 将研究更高的 Reidemeister 挠率不变量（与光滑流形的微分同胚空间相关的代数 K 理论不变量）和与圆丛相关的更简单的循环同调群之间的关系。 特别是，他计划证明 Bokstedt 定理的复杂版本，如果证明的话，将给出这些困难的不变量的明确实现。 他将使用的方法是莫尔斯理论、代数K理论和循环同调。 (3) Jerome P. Levine 将致力于奇维流形的 Atiyah-Patodi-Singer rho 不变量的直接拓扑描述，并确定它何时是同伦不变量。 特别是，他有一个沿着表征路径跳跃的猜想公式。 他还在研究可能使用 rho 不变量（多重签名）的代数版本来确定 Cappell-Shaneson 的同源一致性手术群（Gamma 群）的无扭转部分。 他将继续研究群论中的“代数闭包”构造，特别是其在链接索引问题中的应用。 几个几何问题简化为关于自由群代数闭包的纯代数问题。 (4) Daniel Ruberman 将处理规范论中与低维流形拓扑相关的许多问题。 这项工作的一个主要方面是通过有限能量反自对偶连接的模空间进一步发展有边界的 4 流形规范理论。 应用包括通过低属表面阻碍同源类的表示以及新的结配边不变量。 该项目还包括对 3 流形连接的规范理论不变量的研究，以及这些不变量与平面 SU(2) 连接空间拓扑的关系。 这些不变量给群成为 3 流形的基本群带来了潜在的新障碍。 这些不同的项目涉及流形的拓扑和几何，这些空间局部类似于熟悉的欧几里得空间。 三维和四维流形为空间和时间的物理世界提供了模型，而高维流形自然地作为涉及许多变量的复杂系统的参数空间而出现。 这些空间的基本问题包括它们在数值和代数不变量方面的分类、对称性（或微分同胚）的分类以及一个空间如何位于另一个空间内部的确定（就像一条打结曲线位于 3 维空间内部） 。 布朗正在完善一些所需的代数机器。 Igusa 的项目部分研究了如何用从相应矩阵族导出的代数不变量来理解复杂的对称族。 莱文正在研究解析定义的不变量的拓扑解释，以及它在结理论中的应用。 鲁伯曼将使用数学物理的杨-米尔斯方程来理解四维流形及其包含的二维表面的性质。