The topology of positive scalar curvature

TL;DR

This survey explores the topology of the space of positive scalar curvature metrics on manifolds, highlighting recent advances in index theory, surgery, and smoothing techniques that reveal new obstructions and elements in homotopy groups.

Contribution

It systematically connects large scale index theory and surgery methods to study the existence, classification, and homotopy groups of positive scalar curvature metrics.

Findings

New obstructions from coarse index theory

Construction of infinite order elements in homotopy groups

Non-trivial elements in the moduli space of metrics

Abstract

In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, what are its homotopy groups? Higher index theory of the Dirac operator is the basic tool to address these questions. This has seen tremendous development in recent years, and in this survey we will discuss some of the most pertinent examples. In particular, we will show how advancements of large scale index theory (also called coarse index theory) give rise to new types of obstructions, and provide the tools for a systematic study of the existence and classification problem via the K-theory of C*-algebras. This is part of a program "mapping the topology of positive scalar curvature to analysis". In addition, we will show how advanced surgery theory and smoothing theory can be used to construct the first elements of infinite order in the k-th homotopy groups of the space of metrics of positive scalar curvature for arbitrarily large k. Moreover, these examples are the first ones which remain non-trivial in the moduli space of such metrics.