Positive scalar curvature, K-area and essentialness

TL;DR

This paper explores the relationship between positive scalar curvature obstructions, large scale geometric properties like enlargeability and essentialness, and the Rosenberg index, emphasizing the role of infinite K-area in spin manifolds.

Contribution

It systematically links large scale geometric notions such as infinite K-area to index theoretic obstructions to positive scalar curvature, clarifying their interplay.

Findings

Essentialness and non-vanishing Rosenberg index for manifolds with infinite K-area.

Systematic connection between large scale geometry and index theory.

Clarification of the role of K-homology classes of infinite K-area.

Abstract

The Lichnerowicz formula yields an index theoretic obstruction to positive scalar curvature metrics on closed spin manifolds. The most general form of this obstruction is due to Rosenberg and takes values in the $K$-theory of the group $C^*$-algebra of the fundamental group of the underlying manifold. We give an overview of recent results clarifying the relation of the Rosenberg index to notions from large scale geometry like enlargeability and essentialness. One central topic is the concept of $K$-homology classes of infinite $K$-area. This notion, which in its original form is due to Gromov, is put in a general context and systematically used as a link between geometrically defined large scale properties and index theoretic considerations. In particular, we prove essentialness and the non-vanishing of the Rosenberg index for manifolds of infinite $K$-area.