$K$-theory and homotopies of 2-cocycles on higher-rank graphs

Abstract

This paper continues our investigation into the question of when a homotopy $\omega = \{\omega_t\}_{t \in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, \omega_0)) \cong K_*(C^*(\mathcal{G}, \omega_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_\Lambda$ is the infinite path groupoid associated to a row-finite higher-rank graph $\Lambda$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $\Lambda$, then $K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_0})) \cong K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_1})),$ where $\sigma_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_\Lambda$ associated to the 2-cocycle $c_t$ on $\Lambda$. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semi-continuous $C^*$-bundle.