In the works of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is the starting point for an investigation of the relationships between analytic and topological invariants of smooth orientable manifolds. The signature and related K-theory invariants, primary and secondary, are used to 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 primary 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 showt hat secondary invariants, such as the rho-class, also extend from closed manifolds to Cheeger spaces. We give a rigorous account of 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.

STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR / Albin, P; Piazza, P. - In: ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE. - ISSN 0012-9593. - 55:1(2022), pp. 43-108. [10.24033/asens.2491]

### STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR

### Albin, P;Piazza, P

### 2022

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