Paper · Jul 6, 2025

Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts

Mauricio Angel

math.DGarXiv:2507.04512
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Abstract

We present a comprehensive analysis of Bredon's trick, a powerful local-to-global extension principle with broad applications across differential geometry and computational topology. Our main contributions include: (1) novel applications to stratified pseudomanifolds via Verona cohomology with explicit verification of axiomatic conditions; (2) new frameworks for Ricci flow singularity analysis using local curvature concentration; (3) stability theorems for persistent homology in distributed computational settings; and (4) rigorous applications to medical imaging and neural network topology. By systematically developing the theoretical foundations and providing concrete implementations, this work establishes Bredon's trick as a unifying framework for modern local-to-global arguments in geometric analysis and applied topology.

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Submitted
Jul 6, 2025
Subjects
math.DG

Comments: arXiv admin note: substantial text overlap with arXiv:2008.13279

Ricci flowSingularitiesStability & uniquenessDiscrete & graph Ricci

Its neighbourhood

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.