Fredholm-module representative of the Agranovich–Dynin lift

Construct a concrete Fredholm module over the boundary space that represents the K-homology lift of the difference [D_d]-[D_e] associated with two self-adjoint, local, regular boundary conditions for the same elliptic operator.

Background

For two self-adjoint, local, regular boundary conditions D_d and D_e, the corresponding absolute K-homology classes have the same image in relative K-homology. Consequently, their difference lifts to a K-homology class supported on the boundary.

The classical Agranovich–Dynin formula computes the difference of the Fredholm indices associated with the two boundary conditions. The unresolved problem is to describe the corresponding K-homology lift explicitly by a Fredholm module.

References

Is there a concrete way to describe a Fredholm module over $$ that represents this lift?

— Regular boundary conditions and analytic K-homology  (2610.02899 - Engel et al., 2 Oct 2026) in Section 8, Further questions, item 4 (bullet_AD)

The obvious conjecture here is that the $K$-homological lift sought in \cref{bullet_AD} is exactly given by the class of this operator. Hence: How does one get absolute classes from the Boutet de Monvel calculus?

— Regular boundary conditions and analytic K-homology  (2610.02899 - Engel et al., 2 Oct 2026) in Section 8, Further questions, item 5

Therefore: Can one use $K$-homology and a hypothetical $K$-homological Agranovich--Dynin formula to describe spaces of local, regular boundary conditions? More generally, in \cref{SecNonlocal} we associate to a regular, non-local boundary condition $D_e$ a space $X_e$ so that $D_e$ defines a $K$-homology class over $X_e$. If $D_e$ and $D_d$ are two regular boundary conditions connected by a Riesz-continuous path, how do the spaces $X_e$ and $X_d$ relate? Are they homotopy equivalent to each other? Can these spaces and the resulting $K$-homology classes over them be used to describe spaces of regular, not-necessarily-local boundary conditions?

— Regular boundary conditions and analytic K-homology  (2610.02899 - Engel et al., 2 Oct 2026) in Section 8, Further questions, item 6