Higher rho invariants and the moduli space of positive scalar curvature metrics

Abstract

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group P˜(M) of positive scalar curvature metrics is defined to be the quotient abelian group of this action, i.e. the coinvariant of the action. P˜(M) measures the size of the moduli space of positive scalar curvature metrics on M. In this paper, we use the higher rho invariant and the finite part of the K-theory of the group C⁎-algebra of π1(M) to give a lower bound of the rank of the moduli group P˜(M). The key idea of our proof is the use of higher rho invariant; this is a secondary invariant associated to Dirac operators and in particular depends on the choice of Riemannian metric. We show that the higher rho invariant remains unchanged in a certain K-theory group under the action of the diffeomorphism group, allowing us to distinguish elements in P˜(M).