K-theory and homotopies of 2-cocycles on group bundles

This paper continues the author's program of investigating the question of when a homotopy of 2-cocycles Ω = {ω_t}_t∈[0,1] on a locally compact Hausdorff groupoid G induces an isomorphism of the K-theory groups of the twisted groupoid C∗-algebras: K∗(C∗(G, ω₀)) ≅ K∗(C∗(G, ω₁)). Building on our earlier work in [4, 5], we show that, if π : G → M is a locally trivial bundle of amenable groups over a locally compact Hausdorff space M, a homotopy Ω = {ω_t}_t∈[0,1] of 2-cocycles on G gives rise to an isomorphism: K∗(C∗(G, ω₀)) ≅ K∗(C∗(G, ω₁)).