Relative differential K-theory

Abstract

Let $\rho:Y\rightarrow X$ be a smooth map between two smooth compact manifolds. We define the relative differential K-theory group $\check{K}^*(\rho)$, and show that it fits into a six-term exact sequence. We define $\check{K}^*(\rho,\mathbb{R}/\mathbb{Z})$, the K-theory of $\rho$ with $\mathbb{R}/\mathbb{Z}$ coefficients. It turns out that $\check{K}^*(\rho,\mathbb{R}/\mathbb{Z})$ is isomorphic to the group of homomorphisms from the relative K-homology of $\rho$ to $\mathbb{R}/\mathbb{Z}$ up to a degree-shift by one.