The topology of positive scalar curvature

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.

25 pages, to appear in the proceedings of the ICM 2014 in Seoul; v2: a number of mistakes corrected (thanks to Greg Landweber for pointing some of them out)

References in corpus (2)

• Positive curvature, partial vanishing theorems, and coarse indices
• The Mayer-Vietoris Sequence for the Analytic Structure Group

Cited by in corpus (9)

• Width, Largeness and Index Theory
• Coarse geometry and Callias quantisation
• An $H{P}^2$-bundle over $S^4$ with nontrivial $\hat{A}$-genus
• The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the C*-index
• A Markovian and Roe-algebraic approach to asymptotic expansion in measure
• A note on invertibility of the Dirac operator twisted with Hilbert-A-module coefficients
• Homotopy equivalence of spaces of metrics with invertible Dirac operator
• Strongly Quasi-local algebras and their $K$-theories
• Band width and the Rosenberg index