Linear algebra and unification of geometries in all scales

TL;DR

This paper introduces a unifying framework for small and large scale geometries using multilinear forms, providing new compactifications and simplified proofs of coarse topology theorems.

Contribution

It proposes a novel unification of geometric scales via multilinear forms, encompassing all known compactifications and establishing a new criterion for large scale equivalences.

Findings

Unified small and large scale geometries through multilinear forms.

All major compactifications are special cases of the proposed framework.

A new characterization of large scale equivalences via Higson coronas.

Abstract

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gromov boundary of hyperbolic spaces, the visual boundary of CAT(0)-spaces, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. As an application we get simple proofs of results generalizing well-known theorems from coarse topology. A new result (at least to the author) is the following (see \ref{HomeoOfHigsonImpliesLSEquivalence}):\\ \emph{A coarse bornologous function $f:X\to Y$ of metrizable large scale spaces is a large scale equivalence if and only if it induces a homeomorphism of Higson coronas.} This paper is an extension of \cite{JD2} and, at the same time, it overrides \cite{JD2}.