Stability of almost submetries

Abstract

In this paper, we consider a triple of Gromov-Hausdorff convergence: A_i\stackrel{d_{GH}}{\to }A, B_i\stackrel{d_{GH}}{\to }B 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 ◦ Ψ_i = Φ_i ◦ 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.

Key words： Gromov-Hausdorff convergence; Alexandrov space; stability; fiber bundle; (almost) submetry

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