[Paper Review] A coarse invariant

Addison Fox, Brendon LaBuz+1 more|arXiv (Cornell University)|Dec 7, 2011

Algebraic structures and combinatorial models1 references3 citations

TL;DR

This paper extends a metric space invariant—previously defined under bornologous equivalences—into the coarse category, establishing a coarse invariant that preserves structural properties under large-scale geometric transformations. The key contribution is a robust, invariant framework for comparing spaces in coarse geometry via bornologous maps, enabling classification of spaces up to coarse equivalence.

ABSTRACT

This note extends the invariant defined in An invariant of metric spaces under bornologous equivalences to the coarse category.

Motivation & Objective

• To generalize an existing invariant defined for metric spaces under bornologous equivalences to the broader context of the coarse category.
• To preserve geometric structure under large-scale transformations, enabling classification of spaces up to coarse equivalence.
• To establish a formal framework where the invariant remains unchanged under coarse morphisms, ensuring consistency across coarse-geometric contexts.

Proposed method

• Adapts the original invariant from bornologous equivalences in metric spaces to the setting of the coarse category.
• Uses the category-theoretic structure of the coarse category to define invariance under coarse maps.
• Applies the concept of bornologous maps to ensure that large-scale geometric features are preserved.
• Defines the invariant in terms of equivalence classes of controlled sets, ensuring stability under coarse equivalences.
• Demonstrates that the invariant is well-defined and functorial in the coarse category.
• Relies on the axiomatic framework of coarse structures to generalize the invariant beyond metric spaces.

Experimental results

Research questions

• RQ1 How can an invariant defined for metric spaces under bornologous equivalences be extended to the coarse category?
• RQ2 What properties must the invariant satisfy to remain stable under coarse morphisms?
• RQ3 Is the extended invariant well-defined and functorial in the coarse category?
• RQ4 Can the invariant distinguish between spaces that are not coarsely equivalent?
• RQ5 Does the extension preserve the original invariant's behavior on metric spaces?

Key findings

• The invariant is successfully extended from metric spaces to the coarse category, maintaining its invariance under coarse equivalences.
• The extended invariant is functorial, meaning it respects the morphisms in the coarse category.
• The construction preserves the original invariant's behavior on metric spaces, ensuring consistency.
• The invariant remains stable under large-scale deformations, as required in coarse geometry.
• The framework provides a new tool for classifying spaces in coarse geometry using algebraic invariants.
• The result establishes a bridge between bornologous equivalences and the categorical structure of coarse spaces.