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

TL;DR

This paper extends Bredon's trick, a local-to-global principle, to various geometric and topological contexts, including stratified spaces, Ricci flow, persistent homology, and applications in medical imaging and neural networks.

Contribution

It introduces new applications and frameworks for Bredon's trick in stratified pseudomanifolds, Ricci flow analysis, and distributed persistent homology, with concrete implementations.

Findings

Verified axiomatic conditions for Verona cohomology in stratified pseudomanifolds

Developed local curvature concentration methods for Ricci flow singularities

Established stability theorems for persistent homology in distributed systems

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.