Functoriality for higher rho invariants of elliptic operators

Abstract

Let N be a closed spin manifold with positive scalar curvature and DN the Dirac operator on N . Let M1 and M2 be two Galois covers of N such that M2 is a quotient of M1. Then the quotient map from of M1 to M2 naturally induces maps between the geometric C * -algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the maximal higher rho invariants of the lifts of DN to M1 and M2 behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.