Relative differential K-theory

Abstract

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