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(Stratified) surgery, K-theory and the signature operator

Banff International Research Station for Mathematical Innovation and Discovery

Let $X$ be an orientable smooth manifold without boundary. The surgery sequence associated to $X$, due to Browder, Novikov, Sullivan and Wall, is a fundamental object in differential topology. Browder and Quinn also developed a version of this sequence for smoothly stratified spaces. The goal of this talk is to explain how it is possible to use the signature operator in order to map the surgery sequence in topology to a sequence of K-theory groups for C^*-algebras, called the analytic surgery sequence. The original result is due in the smooth case to Higson and Roe but I will instead explain an alternative approach developed by Schick and myself. I will also explain how, building on joint work with Albin, Leichtnam, Mazzeo it is possible to map the Browder-Quinn sequence associated to a Cheeger space to the analytic surgery sequence. This talk is based on joint work with Thomas Schick and ongoing work, still in progress, with Pierre Albin.

Mathematics; Geometry; Partial differential equations; Differential geometry

video/mp4

Author affiliation: University of Rome `Sapienza'

2017-06-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61940

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/