  We establish existence of the eta-invariant as well as of the
Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac
operators on an incomplete edge space. Our analysis applies in particular to
the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an
analogue of the Atiyah-Patodi-Singer index theorem for incomplete edge spaces
and their non-compact infinite Galois coverings with edge singular boundary.
Our arguments employ microlocal analysis of the heat kernel asymptotics on
incomplete edge spaces and the classical argument of Atiyah-Patodi-Singer. As
an application, we discuss stability results for the two rho-invariants we have
defined.
