  In work of Higson-Roe the fundamental role of the signature as a homotopy and
bordism invariant for oriented manifolds is made manifest in how it and related
secondary invariants define a natural transformation between the
(Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact
sequence of C*-algebra K-theory groups.
  In recent years the (higher) signature invariants have been extended from
closed oriented manifolds to a class of stratified spaces known as L-spaces or
Cheeger spaces. In this paper we show that secondary invariants, such as the
rho-class, also extend from closed manifolds to Cheeger spaces. We revisit a
surgery exact sequence for stratified spaces originally introduced by
Browder-Quinn and obtain a natural transformation analogous to that of
Higson-Roe. We also discuss geometric applications.
