STRATIFIED SURGERY AND K-THEORY INVARIANTS OF THE SIGNATURE OPERATOR

– 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 show that secondary invariants, such as the -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.