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

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This paper continues our investigation into the question of when a homotopy ω= {ωₜ}ₜ ∈ [₀,₁] of 2-cocycles on a locally compact Hausdorff groupoid G gives rise to an isomorphism of the K-theory groups of the twisted groupoid C*-algebras: K*(C*(G, ω₀)) K*(C*(G, ω₁)). In particular, we build on work by Kumjian, Pask, and Sims to show that if G = GΛ is the infinite path groupoid associated to a row-finite higher-rank graph Λ with no sources, and {cₜ}ₜ ∈ [₀,₁] is a homotopy of 2-cocycles on Λ, then K*(C*(GΛ, σc₀)) K*(C*(GΛ, σc₁)), where σcₜ denotes the 2-cocycle on GΛ associated to the 2-cocycle cₜ on Λ. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid G gives rise to an upper semi-continuous C*-bundle.