On the range of the relative higher index and the higher rho-invariant for positive scalar curvature

TL;DR

This paper refines lower bounds on the size of the space of positive scalar curvature metrics on a spin manifold by analyzing the relative higher index map and its relation to the Baum-Connes assembly map.

Contribution

It provides a sharper lower bound for the rank of the positive scalar curvature metric group using the image of the relative higher index map and its connection to K-theory and the Baum-Connes conjecture.

Findings

The relative higher index map's image contains the Baum-Connes assembly map's image rationally.

New lower bounds for the positive scalar curvature bordism group are established.

The bounds depend on the dimension of the manifold and the structure of its fundamental group.

Abstract

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metrics on $M$. Weinberger and Yu gave a lower bound of the rank of $P(M)$ in terms of the number of torsion elements of $\pi_1(M)$. In this paper, we give a sharper lower bound of the rank of $P(M)$ by studying the image of the relative higher index map from $P(M)$ to the real K-theory of the group $\mathrm{C}^\ast$-algebra $\mathrm{C}^\ast_{\mathrm{r}}(\pi_1(M))$. We show that it rationally contains the image of the Baum-Connes assembly map up to a certain homological degree depending on the dimension of $M$. At the same time we obtain lower bounds for the positive scalar curvature bordism group by applying the higher rho-invariant.