Local index theory and $\mathbb{Z}/k\mathbb{Z}$ $K$-theory

Abstract

For any given submersion $\pi:X\to B$ with closed, oriented and spin$^c$ fibers of even dimension, equipped with a Riemannian and differential spin$^c$ structure, we apply the Atiyah-Singer-Gorokhovsky-Lott approach to the local family index theorem without the kernel bundle assumption to construct an analytic index $\textrm{ind}^a_k$ in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory at the cocycle level. This is achieved by associating to every cocycle $(\mathbf{E}, \mathbf{F}, \alpha)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $X$ a cocycle $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, \alpha)$ of the odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory group of $B$. We also prove a Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory, which expresses the Cheeger-Chern-Simons form of $\textrm{ind}^a_k(\mathbf{E}, \mathbf{F}, \alpha)$ in terms of that of $(\mathbf{E}, \mathbf{F}, \alpha)$. Furthermore, we show that the analytic index $\textrm{ind}^a_k$ and the Riemann-Roch-Grothendieck-type formula in odd $\mathbb{Z}/k\mathbb{Z}$ $K$-theory refine the underlying geometric bundle of the analytic index and the Riemann-Roch-Grothendieck theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory, respectively.