For noncompact hypersurfaces satisfying mild geometric conditions, the partitioned index of a Dirac operator equals the index of the induced Dirac operator on the hypersurface.
The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature
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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.
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A partitioned manifold index theorem for noncompact hypersurfaces
For noncompact hypersurfaces satisfying mild geometric conditions, the partitioned index of a Dirac operator equals the index of the induced Dirac operator on the hypersurface.