Paper detail

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

This paper continues our investigation into the question of when a homotopy $ω= \{ω_t\}_{t \in [0,1]}$ 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: $K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_Λ$ is the infinite path groupoid associated to a row-finite higher-rank graph $Λ$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $Λ$, then $K_*(C^*(\mathcal{G}_Λ, σ_{c_0})) \cong K_*(C^*(\mathcal{G}_Λ, σ_{c_1})),$ where $σ_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_Λ$ associated to the 2-cocycle $c_t$ on $Λ$. 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.