Quantitative submetry models for Alexandrov collapse

Definitions and mathematical statement

The source is the open-ended instruction "Study the collapse of Alexandrov spaces." A standard compact formulation considers Alexandrov spaces $X_i$ with a common lower curvature bound, Gromov--Hausdorff convergence $X_i\to Y$, and $\dim Y<\dim X_i$. The source does not fix dimensions, compactness, normalization, boundary, curvature scale or the desired structural conclusion. The supplied aid treats quantitative models admitting exact metric submetries; it does not claim every collapse has such a map.

A submetry $p:X\to Y$ of compact metric spaces is a surjection satisfying \[ p(\overline B_X(x,r))=\overline B_Y(p(x),r)\quad\text{for all }x\in X,\ r\ge0. \] Define its fiber defect $\Delta(p)=\sup_{y\in Y}\operatorname{diam}_X p^{-1}(y)$. The reported graph-correspondence formula is \[ \operatorname{dis}\{(x,p(x)):x\in X\} =\sup_{x,x'}|d_X(x,x')-d_Y(p(x),p(x'))|=\Delta(p), \] so $d_{\rm GH}(X,Y)\le\Delta(p)/2$. For compact submetries $X\xrightarrow pY\xrightarrow qZ$, the aid also reports \[ \Delta(q\circ p)\le\Delta(p)+\Delta(q). \] The equality concerns the distortion of this particular correspondence; it does not claim the Gromov--Hausdorff upper bound is always exact.

For $0<\varepsilon\le1$, take the spherical suspension of the circle of circumference $2\pi\varepsilon$, with suspension coordinate $t\in[0,\pi]$. The result $X_\varepsilon$ is a compact two-dimensional Alexandrov sphere with curvature bounded below by one, with cone singularities at the poles if $\varepsilon<1$. Projection to $t$ is a submetry to $[0,\pi]$. The aid gives its extrinsic fiber-diameter profile \[ \operatorname{diam}_{X_\varepsilon}p^{-1}(t)= \arccos\left(\cos^2t+\sin^2t\cos(\pi\varepsilon)\right), \] with \[ \Delta(p)=\pi\varepsilon,\qquad \operatorname{Area}(X_\varepsilon)=4\pi\varepsilon,\qquad d_{\rm GH}(X_\varepsilon,[0,\pi])\le\frac{\pi\varepsilon}{2}\to0. \] Interior fibers are circles and the two endpoint fibers are points, precisely over the endpoint extremal strata of the interval. The diameter formula uses ambient distances, not just intrinsic lengths on a latitude circle. Thus the model gives collapse with a trivial circle fibration over the open interval, but a single ordinary fiber-bundle model cannot extend over both endpoints. These estimates and a model obstruction do not constitute a classification of general Alexandrov collapse.

Short English statement

Study dimension-lowering Alexandrov limits with explicit hypotheses. The supplied aid quantifies compact submetry collapse by fiber diameter and gives an exact spherical-suspension model in which circle fibers degenerate at endpoint strata, without asserting a universal fibration theorem.

Sources

• Problem 10, AIM workshop Manifolds with nonnegative sectional curvature, https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.
• Local UnsolvedMath registry, permanent ID 20003135, exact code catalogue entry; canonical location aim-topology-notes.json, notes[222], dated 14 August 2026; statement labeled exact.
• Research attempt 1: partial result, problem partially solved at the run, rigor label proved, novelty confidence low; registry status partially solved. The fiber-defect package and suspension model are machine-generated research aids, not peer reviewed; independent verification is required before citation. No separate full authoritative bibliography is supplied.