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The Gromov–Lawson codimension $2$ obstruction to positive scalar curvature and the $C^*$–index

Yosuke Kubota and Thomas Schick

Geometry & Topology 25 (2021) 949–960
Abstract

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that this obstruction also can be obtained from the Rosenberg index of the ambient manifold M, which takes values in the K–theory of the maximal C –algebra of the fundamental group of M, using relative index constructions.

In this note, we give a slightly simplified account of Kubota’s work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick and Xie.

Keywords
higher index theory, positive scalar curvature, higher signature, $C^*$–index theory
Mathematical Subject Classification 2010
Primary: 19K56, 46L80, 58J22
References
Publication
Received: 3 October 2019
Revised: 19 March 2020
Accepted: 19 April 2020
Published: 27 April 2021
Proposed: John Lott
Seconded: David M Fisher, Bruce Kleiner
Authors
Yosuke Kubota
Department of Mathematical Sciences
Shinshu University
Matsumoto
Japan
Thomas Schick
Mathematisches Institut
Universität Göttingen
Göttingen
Germany
http://www.uni-math.gwdg.de/schick