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

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 ${\mathrm{C}}_{\mathrm{r}}^{\ast }({\pi }_1(M))$. 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.

24 pages, 1 figure; v2: added details to Section 3.2 and a new Section 4; v3: Minor revision. To appear in Adv. Math

References in corpus (2)

• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
• Positive scalar curvature and low-degree group homology

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• Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
• Higher rho invariant and delocalized eta invariant at infinity