Stability of almost submetries

Abstract

In this paper, we consider a triple of Gromov-Hausdorff convergence: $A_i ^{\underrightarrow {d_{GH} }} A,B_i ^{\underrightarrow {d_{GH} }} A$ 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.

Keywords

Gromov-Hausdorff convergence / Alexandrov space / stability / fiber bundle / (almost) submetry

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