Index theoretic approaches to the classification of positive scalar curvature

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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.