An Obstruction to Higher-Dimensional Kernel of Dirac Operators

TL;DR

This paper introduces a $K$-theoretic obstruction that limits the dimension of the kernel of Dirac operators in higher dimensions, using topological $K$-theory and Chern classes to analyze the kernel's properties.

Contribution

It presents a novel $K$-theoretic framework to identify obstructions to increasing the kernel dimension of Dirac operators.

Findings

Identifies a $K$-theoretic obstruction to higher kernel dimensions.

Uses Chern classes of $K$-theory classes to analyze Dirac operator kernels.

Provides a topological criterion for kernel dimension limitations.

Abstract

This paper provides a $K$-theoretic obstruction for higher kernel dimension for Dirac operators. For this we use a fibre-wise Dirac operator that gives rise to a family of Fredholm operators representing a class in topological $K$-theory. Then Chern classes of this $K$-class contain some information about the kernel of the operators.