Stability of almost submetries

In this paper, we consider a triple of Gromov-Hausdorff convergence: and maps f_i: A_i → B_i converge to a map f: A → B, where A_i are compact Alexandrov n-spaces and B_i are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψ_i: A_i → A, Φ_i: B_i → B such that f {ring operator} Ψ_i = Φ_i {ring operator} f_i. When f is an ε-submetry with ε > 0, we obtain a sufficient condition for the stability in the case that A_i are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.