Codimension 1 transfer maps of K theoretic indexes

TL;DR

This paper explores transfer maps in K-theory related to spin manifolds and their submanifolds, providing alternative formulations and extending results to higher K-theoretic signatures, with implications for scalar curvature obstructions.

Contribution

It offers an alternative formulation of the codimension 1 transfer map in K-theory and extends the results to higher K-theoretic signatures.

Findings

Transfer map between K groups of group C*-algebras is explicitly constructed.

The Rosenberg index of a submanifold obstructs positive scalar curvature on the ambient manifold.

Extension of transfer results to higher K-theoretic signatures.

Abstract

Let $M$ be a closed spin manifold and $N$ be a codimension 1 submanifold of it. Given certain homotopy conditions, Zeidler shows that the Rosenberg index of $N$ is an obstruction to the existence of positive scalar curvature on $M$. He further gives a transfer map between the K groups of the group $C^*$ algebras of the foundemental group. The transfer map maps the Rosenberg index of $M$ to the one of $N$. In this note, we present an alternative formulation of the transfer map using maps between $C^*$ algebras, and give an analogus result for the codimension 1 transfer of higher K theoretic signatures.