[Paper Review] Homotopy lifting property of an $e^ε$-Lipschitz and co-Lipschitz map

Shicheng Xu|arXiv (Cornell University)|Nov 26, 2012

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper establishes the homotopy lifting property for $e^\epsilon$-Lipschitz and co-Lipschitz maps between metric spaces, particularly in the context of Alexandrov spaces with curvature bounded below. It proves that such maps are Hurewicz fibrations under mild conditions, extending fibration theorems to singular limits in Gromov-Hausdorff convergence and enabling long exact sequences in homotopy and nilpotency results for fundamental groups of almost nonnegatively curved spaces.

ABSTRACT

An $e^ε$-Lipschitz and co-Lipschitz map, as a metric analogue of an $ε$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stability in Gromov-Hausdorff topology. Due to an overlook in the previous version, the part for bounding intrinsic distance of fibers will be talked about in the coming papers.

Motivation & Objective

• To establish the homotopy lifting property for $e^\epsilon$-Lipschitz and co-Lipschitz maps in the context of metric geometry and Alexandrov spaces.
• To extend fibration theorems to singular limits of collapsing sequences of Alexandrov spaces under Gromov-Hausdorff convergence.
• To generalize results on fundamental group structure, particularly nilpotency of subgroups, to almost nonnegatively curved Alexandrov spaces.
• To provide a metric analogue of Riemannian submersions that retains homotopical control despite lack of unique horizontal lifts.

Proposed method

• Uses neighborhood retractions $\varphi_p$ to fibers via iterated gradient deformations of distance functions, replacing unique horizontal lifts.
• Applies local almost isometries via $(n,\delta)$-strainers in Alexandrov spaces to model local geometry near points.
• Employs Gromov-Hausdorff topology to analyze convergence of sequences of Alexandrov spaces with curvature bounded below.
• Relies on the definition of $e^\epsilon$-LcL maps: $B_{e^{-\epsilon}r}(f(p)) \subseteq f(B_r(p)) \subseteq B_{e^{\epsilon}r}(f(p))$ for all $r>0$, ensuring metric control.
• Reduces the global fibration problem to local properties by leveraging the local almost isometry structure of strainer-regular points.
• Applies results from Perelman and Kapovitch-Petrunin-Tueschmann on fundamental group structure in collapsing spaces.

Experimental results

Research questions

• RQ1 Does an $e^\epsilon$-Lipschitz and co-Lipschitz map between metric spaces satisfy the homotopy lifting property?
• RQ2 Can such maps serve as fibrations in the absence of unique horizontal lifts, as in Riemannian submersions?
• RQ3 To what extent do fibration theorems for Riemannian submersions extend to singular limits of Alexandrov spaces?
• RQ4 What homotopical and fundamental group structures are preserved under Gromov-Hausdorff limits of collapsing Alexandrov spaces?
• RQ5 Can nilpotency of the fundamental group be uniformly bounded in almost nonnegatively curved Alexandrov spaces?

Key findings

• A proper $(1.023)$-LcL map $f: X \to B$ from a finite-dimensional Alexandrov space $X$ with curvature bounded below to an $n$-dimensional Riemannian manifold $B$ is a Hurewicz fibration.
• For any $n$, there exists $\delta_0(n)$ such that if $B$ is $n$-dimensional and each point is $(n,\delta)$-strained with $\delta < \delta_0(n)$, then $f$ is a Hurewicz fibration.
• The homotopy fiber of such maps admits a long exact sequence in homotopy groups, with $f_*$ inducing isomorphisms $\pi_l(X_i, F_i, x_i) \to \pi_l(X, x)$ for $l \geq 1$.
• The fundamental group of the homotopy fiber contains a nilpotent subgroup of index bounded by a universal constant $C(n)$ depending only on dimension $n$.
• The result extends to almost nonnegatively curved Alexandrov spaces: if $\operatorname{diam}(M)^2 \cdot \kappa > -\epsilon(n)$, then $\pi_1(M)$ has a nilpotent subgroup of index $\leq C(n)$.
• The fibration structure persists under Gromov-Hausdorff limits of sequences of Alexandrov spaces with curvature bounded below, even when the limit has weak singularities.