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

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