Stratified surgery and K-theory invariants of the signature operator

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.