K-theory and homotopies of 2-cocycles on transformation groups

This paper constitutes a first step in the author's program to investigate the question of when a homotopy of 2-cocycles w = {w_t}_tε[0,1] on a locally compact Hausdorff groupoid G induces an isomorphism of the Ktheory groups of the reduced twisted groupoid C*-algebras: Generalizing work of S. Echterhoff, W. Lück, N.C. Phillips, S. Walters, J. Reine Angew. Math. 639(2010), 173-221, we show that if G = G ⋉ X is a transformation group such that G satisfies the Baum-Connes conjecture with coefficients, a homotopy w = {w_t}_tε[0,1] of 2-cocycles on G ⋉ X gives rise to an isomorphism.