Volume 25, issue 2 (2021)

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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 $M\phantom{\rule{-0.17em}{0ex}}$, 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 $M\phantom{\rule{-0.17em}{0ex}}$, which takes values in the K–theory of the maximal ${C}^{\ast }$–algebra of the fundamental group of $M\phantom{\rule{-0.17em}{0ex}}$, 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
Publication
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