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$λ$-ring structure in differential K-theory

Bo Liu, Xiaonan Ma

TL;DR

This work proves a splitting principle for differential K-theory and shows that the differential $K^0$-ring of a closed manifold carries a natural $\lambda$-ring structure. It then provides a constructive realization of Adams operations in differential K-theory, compatible with both the curvature map and the underlying Chern character, and extends these results to a $T_g$-equivariant setting for compact Lie group actions. The framework unifies algebraic $\lambda$-ring techniques with differential refinements, enabling explicit exterior powers and Adams operations in differential contexts. Overall, the results deliver a concrete differential-operator realization of classical $\lambda$-ring machinery with equivariant generalizations.

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 $λ$-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.

$λ$-ring structure in differential K-theory

TL;DR

This work proves a splitting principle for differential K-theory and shows that the differential -ring of a closed manifold carries a natural -ring structure. It then provides a constructive realization of Adams operations in differential K-theory, compatible with both the curvature map and the underlying Chern character, and extends these results to a -equivariant setting for compact Lie group actions. The framework unifies algebraic -ring techniques with differential refinements, enabling explicit exterior powers and Adams operations in differential contexts. Overall, the results deliver a concrete differential-operator realization of classical -ring machinery with equivariant generalizations.

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 -ring associated to closed smooth manifolds admits a -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.
Paper Structure (6 sections, 9 theorems, 107 equations)

This paper contains 6 sections, 9 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. $\lambda$-ring structure
  3. Differential K-theory
  4. Proof of Theorem \ref{['thm:1.09']}
  5. Adams operations
  6. $T_g$-equivariant differential K-theory

Key Result

Theorem 1.3

Let $R$ be a pre-$\lambda$-ring. We assume that $R$ is torsion free, that is, for any $r\in R$, $r\neq 0$ and any $n\in \mathbb{N}^*$, $nr=\underbrace{r+\cdots +r}_n \neq 0$. Let $\Psi^n:R\to R$, $n\in \mathbb{N}^*$ be the Adams operations defined by Suppose that for any $n, m\in \mathbb{N}^*$, $a, b\in R$, we have Then the pre-$\lambda$-ring structure of $R$ is a $\lambda$-ring structure.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 3.1
  • ...and 9 more