  We apply the geometric analog of the analytic surgery group of Higson and Roe
to the relative $\eta$-invariant. In particular, by solving a Baum-Douglas type
index problem, we give a "geometric" proof of a result of Keswani regarding the
homotopy invariance of relative $\eta$-invariants. The starting point for this
work is our previous constructions in "Realizing the analytic surgery group of
Higson and Roe geometrically, Part I: The geometric model" (arXiv:1308.5990).

K-Theory and Homology;Operator Algebras