The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature

TL;DR

This paper develops a coarse relative index theory and uses it to create a framework for detecting obstructions to uniform positive scalar curvature via submanifolds, extending Gromov and Lawson's ideas.

Contribution

It introduces a coarse relative index and a method to construct K-theory maps linking manifold and submanifold indices, providing new tools for scalar curvature obstructions.

Findings

Established properties of the coarse relative index.

Constructed a machinery for submanifold obstructions to scalar curvature.

Linked coarse index classes of manifolds and submanifolds via K-theory maps.

Abstract

We provide a coarse version of the relative index of Gromov and Lawson and thoroughly establish all of its basic properties. As an application, we discuss a general procedure to construct wrong way maps on the $K$-theory of the Roe algebra mapping the coarse index class of the Dirac operator of a manifold to the one of a suitably embedded submanifold of arbitrary codimension, thereby establishing an abstract machinery to find obstructions to uniform positive scalar curvature coming from these submanifolds.