General validity of Roe’s pairing equality

Determine whether, for every proper metric space M, every class ξ in algebraic K-theory K_*^{alg}(B M) of the algebra B M of locally trace-class, finite propagation operators on M, and every class x in K^{*-1}(∂_h M) of the Higson corona ∂_h M, the equality ⟨χ ch(ξ), T ch(x)⟩ = ⟨ι_* ξ, x⟩ holds, where χ is the coarse character map from periodic cyclic homology of B M to coarse homology of M, ch is the Chern character from algebraic K-theory to periodic cyclic homology, T is the transgression from Alexander–Spanier cohomology of ∂_h M to coarse cohomology of M, and ι_* is the canonical comparison map from K_*^{alg}(B M) to K_*^{top}(C^*M) for the Roe algebra C^*M.

Background

The paper studies two ways to pair K-theory classes associated to a proper metric space M: one via the coarse character map χ and the algebraic Chern character ch landing in coarse homology, and another via the index pairing between K_^{top}(C^*M) and K^{-1}(∂_h M). Roe established equality of these pairings under specific geometric conditions (e.g., Dirac operators and particular coronas), and subsequent work extended parts of this. The authors formulate the general equality as Equation (PairingEquation) and seek conditions ensuring its validity.

They prove several conditional results (Theorems A–C) showing that Equation (PairingEquation) holds under assumptions such as surjectivity of the analytic assembly map and injectivity of certain comparison maps, with concrete verification for M = R^n. Nonetheless, the question of whether the equality holds in full generality remains unresolved.