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Wall singularity of spaces with an upper curvature bound

Koichi Nagano

TL;DR

This work analyzes wall singularities in GCBA spaces with an upper curvature bound, introducing a codimension-one geometric structure theorem and a codimension-two regularity criterion. The authors develop a comprehensive framework built on strainer maps, spherical suspensions, and almost-spherical structures to decompose and understand singular strata, including relaxations that enable robust, perturbation-tolerant results. They prove a wall singularity theorem describing local product structure near $n$-wall points and provide a geometric characterization of codimension-two regularity, along with a suite of applications such as sphere theorems for CAT$(1)$ homology manifolds and asymptotic regularity for CAT$(0)$ spaces. The methodology combines convexity arguments for distance-coordinate maps, fiber analysis of strainer maps, and ultralimit techniques to derive rectifiability, topological regularity, and stability results across scales. These results advance the understanding of how curvature upper bounds shape the local and asymptotic topology of GCBA spaces, with implications for volume growth and topological rigidity in metric geometry.

Abstract

We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.

Wall singularity of spaces with an upper curvature bound

TL;DR

This work analyzes wall singularities in GCBA spaces with an upper curvature bound, introducing a codimension-one geometric structure theorem and a codimension-two regularity criterion. The authors develop a comprehensive framework built on strainer maps, spherical suspensions, and almost-spherical structures to decompose and understand singular strata, including relaxations that enable robust, perturbation-tolerant results. They prove a wall singularity theorem describing local product structure near -wall points and provide a geometric characterization of codimension-two regularity, along with a suite of applications such as sphere theorems for CAT homology manifolds and asymptotic regularity for CAT spaces. The methodology combines convexity arguments for distance-coordinate maps, fiber analysis of strainer maps, and ultralimit techniques to derive rectifiability, topological regularity, and stability results across scales. These results advance the understanding of how curvature upper bounds shape the local and asymptotic topology of GCBA spaces, with implications for volume growth and topological rigidity in metric geometry.

Abstract

We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.
Paper Structure (66 sections, 83 theorems, 143 equations)

This paper contains 66 sections, 83 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Wall singularity
  4. Relaxed wall singularity
  5. Applications
  6. Sketch of the proof of the wall singularity theorem
  7. Outline of this paper
  8. Preliminaries
  9. Metric spaces
  10. Maps between metric spaces
  11. Geodesic metric spaces
  12. Convergences of metric spaces
  13. CAT$\boldsymbol{(\kappa)}$ spaces
  14. Geodesically complete CAT$\boldsymbol{(\kappa)}$ spaces
  15. Dimension of CAT$\boldsymbol{(\kappa)}$ spaces
  16. ...and 51 more sections

Key Result

Theorem 1.1

Let $X$ be a $\operatorname{GCBA}$ space. Then for every $n$-wall point $x \in W_n(X^n)$ in $X^n$, and for every open neighborhood $U$ of $x$, there exist a point $x_0 \in W_n(X^n)$ arbitrarily close to $x$, and an open neighborhood $U_0$ of $x_0$ contained in $U$, such that $x_0 \in S(U_0)$, and $U

Theorems & Definitions (181)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Lemma 2.2
  • ...and 171 more