Every finite-dimensional analytic space is -homogeneous

All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results: (1) Every finite-dimensional analytic space is -homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is -homogeneous with pairwise disjoint ¹₂ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding -homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is -homogeneous. We also investigate finite unions of homogeneous spaces.