[Paper Review] Hurewicz fibration Theorem and Margulis lemma on Alexandrov spaces

Shicheng Xu, Xuchao Yao|arXiv (Cornell University)|Feb 28, 2019

Advanced Topology and Set Theory23 references4 citations

TL;DR

This paper establishes that regular almost Lipschitz submersions on collapsed Alexandrov spaces with lower curvature bounds are Hurewicz fibrations, and any two such fibrations are homotopy equivalent. As a key application, it proves a uniform index bound version of the generalized Margulis lemma, showing that small loops generate a subgroup containing a nilpotent subgroup of index at most $ w(n) $, depending only on dimension $ n $.

ABSTRACT

In this paper we prove any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration (Theorem A), and any two such fibrations on one collapsed Alexandrov space are homotopy equivalent to each other (Theorem B). As an partial application of Theorem A, we give a proof of the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of $1$-ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$.

Motivation & Objective

• To establish the Hurewicz fibration property for regular almost Lipschitz submersions on collapsed Alexandrov spaces with curvature bounded below.
• To show that any two such fibrations on the same collapsed Alexandrov space are homotopy equivalent.
• To apply the fibration structure to derive a uniform index bound in the generalized Margulis lemma for Alexandrov $ n $-spaces.
• To prove that the index bound depends only on the dimension $ n $, not on the specific point or space.

Proposed method

• Utilizes Yamaguchi's construction of regular almost Lipschitz submersions on collapsed Alexandrov spaces with lower curvature bounds.
• Applies the theory of Hurewicz fibrations to show that these submersions satisfy the homotopy lifting property.
• Employs homotopy equivalence techniques to compare different fibrations on the same space.
• Applies the fibration structure to analyze the fundamental group of small balls in the space.
• Uses the nilpotent subgroup structure of the fundamental group of small balls to derive the uniform index bound.
• Relies on geometric and topological properties of Alexandrov spaces with curvature bounded below to control the group-theoretic behavior of small loops.

Experimental results

Research questions

• RQ1 Is a regular almost Lipschitz submersion on a collapsed Alexandrov space with lower curvature bound a Hurewicz fibration?
• RQ2 Are any two such fibrations on the same collapsed Alexandrov space homotopy equivalent?
• RQ3 Can the fibration structure be used to establish a uniform index bound in the generalized Margulis lemma?
• RQ4 Does the index of the nilpotent subgroup in the fundamental group of a 1-ball depend only on the dimension $ n $?
• RQ5 What is the nature of the fundamental group of small balls in collapsed Alexandrov spaces with curvature bounded below?

Key findings

• Any regular almost Lipschitz submersion on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration.
• All such fibrations on a fixed collapsed Alexandrov space are homotopy equivalent to each other.
• The generalized Margulis lemma holds with a uniform index bound: small loops at any point generate a subgroup containing a nilpotent subgroup of index at most $ w(n) $.
• The constant $ w(n) $ depends only on the dimension $ n $, not on the specific space or point.
• The fibration structure provides a topological framework to analyze the fundamental group of small balls in collapsed Alexandrov spaces.
• The result confirms a uniform geometric-topological control on the local fundamental group structure in non-collapsing limits of Alexandrov spaces.