Stability of almost submetries (Q2430826)

The authors consider a triple of Gromov-Hausdorff convergence: \({A_i}^{\underrightarrow{d_{GH}}}A,\,{B_i}^{\underrightarrow{d_{GH}}}B\) and maps \( f_i :A_i \rightarrow B_i\) converging to a map \( f :A \rightarrow B \), the 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, they give a necessary and sufficient condition for the sequence to be stable, that is, for \(i\) large, there are homeomorphisms \(\Psi_i : A_i \rightarrow A\), \(\Phi_i : B_i \rightarrow B \) such that \(f\circ \Psi_i = \Phi_i\circ f_i\). When \(f\) is an \(\varepsilon\)-submetry with \(\varepsilon >0\), they obtain a sufficient condition for the stability in the case that the \(A_i\) are Riemannian manifolds. Their results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.