Index theoretic approaches to the classification of positive scalar curvature

在这个项目的第一部分，我们将研究一个给定的空间X的哪些同调类可以用“流形”来表示的问题，这些流形允许具有正数量曲率的度量。动机是这样一个事实，即对于特殊的X（= Bp），正标量曲率可以从一个代表的同调类转移到另一个。这回答流形本身承认度量与积极的标量曲率。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画面，我们将因此与奇异空间表示同调类。我们最初的目标将是发展一个合适的奇异边理论，并建立其性质（正）标量曲率。这将被用来确定“标量积极的一部分”的同调分类空间的有限阿贝尔群，除了所谓的“toral”类。在项目的第2部分中，我们将研究存在多少（一致性类）这样的指标的问题。特别是，我们将研究在何种程度上已知的不变量，如（更高）r-不变量，可以用于无挠基本群。

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.