An Obstruction to Higher-Dimensional Kernel of Dirac Operators

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.