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

TL;DR

This paper develops analytic formulas for the topological degree of non-smooth mappings on the boundary of pseudo-convex domains using index theory and Toeplitz operators, extending degree concepts to less regular maps.

Contribution

It introduces new analytic integral formulas for the topological degree of H"older continuous mappings via index theory in non-commutative geometry.

Findings

Derived explicit index formulas for Toeplitz operators with H"older symbols

Provided analytic degree formulas for non-smooth boundary mappings

Extended topological degree concepts to non-smooth, odd-dimensional cases

Abstract

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.