Mathematical Sciences: Topological Index for Proper Actions, Asymptotic Homomorphisms and Equivariant E-Theory

数学科学：适当作用的拓扑索引、渐近同态和等变 E 理论

• 批准号: 9706767
• 负责人: John Trout
• 金额: $ 7.87万
• 依托单位: Dartmouth College
• 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: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706767&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Index; Proper

Abstract Trout The proposed research involves defining an equivariant ``Atiyah-Singer''-type topological index for elliptic differential operators which are invariant with respect to the proper and cocompact action of a locally compact group on the underlying manifold, i.e. this index arises from equivariantly embedding the base manifold into a (possibly infinite dimensional) Euclidean representation of the group and applying an appropriate version of equivariant Bott periodicity. The proof of the corresponding equivariant index theorem, namely that this topological index is equal to G. Kasparov's equivariant analytic index for the elliptic operator, requires the use of equivariant asymptotic homomorphisms of C*-algebras and the associated equivariant version of the E-theory groups of A. Connes and N. Higson. This research has applications to the Baum-Connes Conjecture and the Novikov Conjecture. The proposed research deals with the index theory of elliptic differential operators on manifolds which commute with the action of a locally compact transformation group, i.e., studying the spaces of solutions of differential equations which exhibit certain degrees of internal symmetry. Associated with each elliptic operator is an "analytic index," which can carry important topological and geometric information about the manifold the operator is associated with. However, this analytic index is very difficult to compute in general, requiring a great deal of detailed analysis. I propose to construct, in a more topological and geometric fashion, another method for computing this index, generalizing the case of compact groups considered by Atiyah and Singer. Being able to compute the indices of these elliptic operators should lead to a clearer understanding of a variety of problems in geometry and representation theory, which are important to physics. In particular the project will have a bearing on the Baum-Connes Conjecture and the still-unsolved Novikov Conjecture.

摘要 Trout 提出的研究涉及为椭圆微分算子定义等变“Atiyah-Singer”型拓扑指数，该指数对于基础流形上局部紧群的真和协紧作用是不变的，即该指数来自于将基流形等变嵌入到群的（可能是无限维）欧几里德表示中，并应用等变 Bott 周期的适当版本。相应的等变指数定理的证明，即该拓扑指数等于 G. Kasparov 椭圆算子的等变解析指数，需要使用 C* 代数的等变渐近同态和 E 理论的相关等变版本A. Connes 和 N. Higson 小组。这项研究适用于鲍姆-康尼斯猜想和诺维科夫猜想。 所提出的研究涉及流形上的椭圆微分算子的指数理论，该流形在局部紧变换群的作用下进行交换，即研究表现出一定程度的内部对称性的微分方程的解空间。与每个椭圆算子相关联的是一个“解析索引”，它可以携带与该算子相关联的流形的重要拓扑和几何信息。然而，这个解析指标一般来说计算起来非常困难，需要大量的详细分析。我建议以更拓扑和几何的方式构造另一种计算该索引的方法，概括 Atiyah 和 Singer 考虑的紧群的情况。能够计算这些椭圆算子的指数应该可以使我们更清楚地理解几何和表示论中的各种问题，这对物理学很重要。 特别是该项目将对鲍姆-康尼斯猜想和尚未解决的诺维科夫猜想产生影响。