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

Abstract

This paper continues the author's program to investigate the question of when a homotopy of 2-cocycles $\Omega =\{{\omega }_t{\}}_{t\in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^{\ast }$-algebras: $K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_0))\cong K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_1)).$ Building on our earlier work, we show that if $\pi :\mathcal{G}\to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $\Omega =\{{\omega }_t{\}}_{t\in [0,1]}$ of 2-cocycles on $\mathcal{G}$ gives rise to an isomorphism $K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_0))\cong K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_1)).$

arXiv commentThis version to be published in the Rocky Mountain Journal of Mathematics