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

TL;DR

This paper derives analytic integral formulas for the topological degree of non-smooth, H"older continuous mappings between even-dimensional manifolds using index theory and non-commutative geometry.

Contribution

It extends previous work by providing explicit index-based formulas for the degree in the even-dimensional case, employing pseudo-differential operators and non-commutative geometry techniques.

Findings

Derived explicit index formulas for the degree of non-smooth mappings.

Connected topological degree with index theory of pseudo-differential operators.

Extended the analytic approach to even-dimensional manifolds.

Abstract

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a H\"older continuous vector bundle. The index formula gives an analytic formula for the degree of a H\"older continuous mapping between even-dimensional manifolds. The paper is an independent continuation of the paper "Analytic formulas for topological degree of non-smooth mappings: the odd-dimensional case".