The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the C*-index

TL;DR

This paper discusses a simplified approach to the Gromov-Lawson codimension 2 obstruction to positive scalar curvature, connecting it with the Rosenberg index and extending it to the signature operator, highlighting its homotopy invariance.

Contribution

It simplifies Kubota's work on the codimension 2 index obstruction and extends the framework to the signature operator, emphasizing homotopy invariance of higher signatures.

Findings

Obstruction can be derived from the Rosenberg index in the maximal C*-algebra.

The approach applies to the signature operator, recovering homotopy invariance.

Simplifies previous constructions of the codimension 2 index obstruction.

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.