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 also this obstruction 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, Xie.

Width, Largeness and Index Theory

“…The situation of Example 5.10 was previously studied by Hanke, Pape, and Schick [18]. Moreover, Kubota [23,24] showed that if X is closed, then it has non-vanishing Rosenberg index.…”