  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.

K-Theory and Homology