The relative Mishchenko--Fomenko higher index and almost flat bundles II: Almost flat index pairing

This is the second part of a series of papers which bridges the Chang–Weinberger–Yu relative higher index and geometry of almost flat hermitian vector bundles on manifolds with boundary. In this paper we apply the description of the relative higher index given in Part I to provide the relative version of the Hanke–Schick theorem, which relates the relative higher index with index pairing of a K-homology cycle with almost flat relative vector bundles. We also deal with the quantitative version and the dual problem of this theorem.