Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a Toeplitz operator with H"older continuous symbol. The index formula gives an analytic formula for the degree of a H"older continuous mapping from the boundary of a strictly pseudo-convex domain.

💡 Research Summary

The paper addresses the problem of defining and computing the topological degree of continuous (but not necessarily smooth) maps from the boundary of a relatively compact strictly pseudoconvex domain Ω in a Stein manifold to a target manifold Y, focusing on the odd‑dimensional case. The authors exploit the index theory of Toeplitz operators acting on the Hardy space H²(∂Ω) to translate the degree problem into an analytic index problem.

First, the geometric setting is described. The boundary ∂Ω carries a natural CR structure and admits a Szegő projection P onto H²(∂Ω). For a Hölder‑continuous symbol f∈C^α(∂Ω) (0<α≤1) the Toeplitz operator T_f = P M_f|_{H²} is defined, where M_f denotes multiplication by f. Classical results guarantee that T_f is a Fredholm operator whenever f is invertible in the C^α‑algebra, and its Fredholm index is an integer that can be related to topological data.

The core of the work lies in applying the machinery of non‑commutative geometry, in particular Connes’ Chern‑Connes pairing between K‑theory and cyclic cohomology. The authors construct a (2k+1)‑dimensional cyclic cocycle ω_f associated with the symbol f by using the Szegő kernel S(z,w) and its Bergman‑type relatives. This cocycle implements the dimension‑raising (or “suspension”) map in cyclic cohomology, which is essential for handling odd‑dimensional situations.

By evaluating the Chern character of the class