Index theoretic approaches to the classification of positive scalar curvature

正标量曲率分类的索引理论方法

• 批准号: 5406956
• 负责人: Professor Dr. Thomas Schick
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2007-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5406956?language=en
• 关键词: Index; theoretic; approaches; classification; positive

In Part 1 of this project, we will study the question which homology classes of a given space X can be represented by "manifolds" which admit metrics with positive scalar curvature. Motivation is the fact that, for special X(= Bp), positive scalar curvature can be transferred from one representative of a homology class to another one. This answers which manifolds themselves admit a metric with positive scalar curvature. Not all homology classes can be represented by smooth manifolds. To get a complete picture, we will therefore work with singular spaces representing homology classes. Our initial goal will be to develop a suitable singular bordism theory and establish its properties with respect to (positive) scalar curvature. This shall then be used to determine the "scalar positive part" of the homology of classifying spaces of finite abelian groups, apart from so called "toral" classes. In Part 2 of the project, we will study the question how many (concordance classes of) such metrics exist. In particular, we will investigate to which extent the known invariants for this question, like (higher) r-invariants, can be used for torsion-free fundamental groups.

在本课题的第一部分中，我们将研究给定空间X的哪些同调类可以用允许度量具有正标量曲率的“流形”表示的问题。动机是，对于特殊的X(= Bp)，正标量曲率可以从一个同调类的代表转移到另一个代表。这就回答了哪些流形本身允许度规具有正的标量曲率。并不是所有的同调类都可以用光滑流形表示。为了得到一个完整的图像，我们将使用表示同调类的奇异空间。我们最初的目标将是发展一个合适的奇异边界理论，并建立其关于（正）标量曲率的性质。然后，这将用于确定有限阿贝尔群的分类空间同调的“标量正部分”，除了所谓的“总数”类。在该项目的第2部分中，我们将研究这样的度量存在多少（一致性类）的问题。特别地，我们将研究这个问题的已知不变量，如（更高）r不变量，在多大程度上可以用于无扭转基本群。