When does the Thom class lift through the coarse co-assembly map away from a submanifold?

Determine conditions under which the Thom class of a submanifold N with K-oriented normal bundle, viewed as an element of K(A(N \Subset M; Cl_{0,r})), lies in the image of the coarse co-assembly map away from N, namely \mu_N: K(sHigCor_N(M; Cl_{1,r})) → K((N \Subset M; Cl_{0,r})).

Background

The authors build an abstract machinery producing wrong-way maps on K-theory of Roe algebras, mapping the coarse index on M to the index on a suitably embedded submanifold N. This machinery hinges on representing the Thom class of the normal bundle via a class in the stable Higson corona and lifting it through a coarse co-assembly map away from N. They show this can be done for multi-partitioned manifolds but seek general criteria ensuring such a lift exists, which would yield broad submanifold obstructions to uniform positive scalar curvature away from N.

References

In order to get this abstract machinery started, the important question is as follows. Under which conditions does the Thom class lie in the image of the coarse co-assembly map away from $N$?

— The relative index in coarse index theory and submanifold obstructions to uniform positive scalar curvature (2506.14301 - Engel et al., 17 Jun 2025) in Introduction, Question (labelled Question \ref{question})