$\lambda$-ring structure in differential K-theory

TL;DR

This paper proves that differential K-theory on smooth manifolds has a λ-ring structure, enabling the construction of Adams operations, and extends these results to equivariant settings with Lie group actions.

Contribution

It establishes a λ-ring structure in differential K-theory and extends the theory to equivariant cases, providing new tools for geometric and topological analysis.

Findings

Proves the splitting principle for differential K-theory.

Shows differential K^0 admits a λ-ring structure.

Constructs Adams operations in differential K-theory.

Abstract

We establish the splitting principle for differential K-theory, a refinement of topological K-theory that incorporates geometric data via differential forms. Using this principle, we prove that the differential $K^0$-ring associated to closed smooth manifolds admits a $\lambda$-ring structure. This structure enables a concrete construction of the Adams operations in differential K-theory introduced by Bunke. At last, we extend all these results to an equivariant setting associated with a compact Lie group action.