Non-triviality of the chiral absolute K-homology class

Determine whether the even-dimensional absolute K-homology class defined by the off-diagonal operator formed from the chiral boundary conditions on an odd-dimensional spin manifold is non-trivial, and, if it is non-trivial, determine whether its lift to K_1(D^*(M)) can be used to study metrics of positive scalar curvature and non-negative mean curvature on the boundary.

Background

For an odd-dimensional spin manifold with boundary, the paper constructs an off-diagonal operator from the two chiral boundary conditions and obtains an absolute K-homology class in K_0(M). In the compact case, the index of this operator is zero, so the index does not determine whether the K-homology class itself vanishes.

Under positive scalar curvature and non-negative mean curvature on the boundary, the associated boundary condition is coercive and the K-homology class admits a lift to K_1(D*(M)). The authors suggest that a non-trivial lift could play a role analogous to a higher rho-invariant.

References

Hence, it is natural to ask whether the entire $K$-homology class offdiag_chiral is zero. If it is actually non-trivial, this could have interesting consequences for the study of positive-scalar-curvature metrics on $$. Specifically, if $$ has positive scalar curvature and $$ non-negative mean curvature, this boundary condition is coercive, whence its $K$-homology class admits a lift to $K_1(D*())$ (see \cref{ex_chiral_lift}). This lift would provide an analogue to the higher $\rho$-invariant in the boundaryless case. Since the higher $\rho$-invariant contains much information about the geometry of positive-scalar-curvature metrics (see e.g. ), it would be quite desirable to have a counterpart for manifolds with boundary. Hence, we ask: Is the class offdiag_chiral non-trivial? If so, can the associated lift be used to study metrics of positive scalar curvature and non-negative mean curvature on the boundary?

offdiag_chiral:

[(0D−D+0)]  ∈  K0().\left[ \begin{pmatrix} 0 & D_- \\ D_+ & 0 \end{pmatrix} \right] \; \in \; K_0().

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

While we do not doubt the existence of non-trivial, interesting examples, we currently do not have any. Hence: What are interesting operators with suitable boundary conditions that gives rise to interesting absolute K-homology classes?

— Regular boundary conditions and analytic K-homology  (2610.02899 - Engel et al., 2 Oct 2026) in Section 8, Further questions, item 3