Euler characteristic K-homology classes for manifolds with boundary

Determine whether, for manifolds with boundary, the absolute and relative K-homology classes of the Euler characteristic operator, [D^\chi_{abs}] and [D^\chi_{rel}], lie in the image of the inclusion of a point and whether these classes vanish in the non-compact case.

Background

The paper compares the Euler characteristic operator on closed, boundaryless, and bounded manifolds. On a closed connected even-dimensional manifold, its K-homology class is determined by the Euler characteristic, while on a non-compact manifold without boundary the class vanishes by a result of Rosenberg. For manifolds with boundary, the authors ask whether analogous simplifications hold for the absolute and relative boundary conditions.

The unresolved issue is whether [D^\chi_{abs}] and [D^\chi_{rel}] contain information beyond their indices and whether both classes vanish in the non-compact setting.