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 π1(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 C⁎-algebra Cr⁎(π1(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.