Index theoretic approaches to the classification of positive scalar curvature

在这个项目的第一部分，我们将研究给定空间X的哪些同调类可以由具有正标量曲率的度量的“流形”来表示。动机是，对于特殊的X(=BP)，正的标量曲率可以从一个同调类的一个表示转移到另一个同调类的表示。这就回答了流形本身具有正的标量曲率的度量。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景，因此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，并建立它关于(正)标量曲率的性质。然后，这将被用来确定有限阿贝尔群的分类空间的同调的“标量正部分”，而不是所谓的“类”。在项目的第2部分中，我们将研究这样的度量存在的数量(一致性类)的问题。特别是，我们将调查这个问题的已知不变量，如(更高的)r-不变量，在多大程度上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.