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

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 $C^{\ast }$–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.