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：本文提出的研究涉及到对下流形上局部紧致群的固有紧致作用不变的椭圆微分算子的等变“Atiyah-Singer”型拓扑指标的定义，即该指标是通过将基流形等变嵌入到群的欧几里得表示（可能是无限维的）中并应用适当版本的等变Bott周期性而产生的。相应的等变指标定理的证明，即该拓扑指标等于G. Kasparov对椭圆算子的等变解析指标，需要使用C*-代数的等变渐近同态以及A. Connes和N. Higson的e -理论群的相关等变版本。本研究对Baum-Connes猜想和Novikov猜想也有应用。本文研究了流形上具有局部紧变换群可交换作用的椭圆微分算子的指标理论，即研究了具有一定程度内对称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”，它可以携带有关该算子所关联的流形的重要拓扑和几何信息。然而，这个分析指标通常很难计算，需要进行大量的详细分析。我建议以更拓扑和几何的方式构造另一种计算该指标的方法，推广Atiyah和Singer所考虑的紧群的情况。能够计算这些椭圆算子的指标应该会导致对几何和表示理论中各种问题的更清晰的理解，这对物理学很重要。特别是，该项目将对鲍姆-康内斯猜想和尚未解决的诺维科夫猜想产生影响。