Higson compactification and dimension raising

Austin, Kyle; Virk, Žiga

doi:10.1016/j.topol.2016.10.005

Cited by 7 publications

(11 citation statements)

References 18 publications

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“…Note that for proper metric spaces, condition (2) in the above proposition follows from Proposition 2.3 in [7] and plays a crucial role in relating properties of a proper metric space with its Higson corona in various places in the literature (see for example [3] or [7]),…”

Section: The Higson Compactification and Coronamentioning

confidence: 99%

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

Dydak

Weighill

2018

Mediterr. J. Math.

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We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.

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Section: The Higson Compactification and Coronamentioning

confidence: 99%

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

Dydak

Weighill

2018

Mediterr. J. Math.

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“…The goal of this section is to give simple proofs of results that generalize the work of Austin-Virk [1] on Dimension Raising maps in coarse category.…”

Section: Dimension Of Formed Setsmentioning

confidence: 99%

“…Case 1: First consider the case of X and Y being metrizable. It is known (see [1]) that asdim(Y ) is finite. By 14.3 it suffices to show dim(∂(ω Y )) ≤ dim(∂(ω X ))+ n − 1.…”

Section: Dimension Of Formed Setsmentioning

confidence: 99%

Linear algebra and unification of geometries in all scales

Dydak

2019

Preprint

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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, Čech-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 16.7): A coarse bornologous function f : X → 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 [7] and, at the same time, it overrides [7].

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“…The following lemma is a special case of Proposition 2.3 from [11]. The proof of the following theorem was inspired in part by the techniques used in [3]. Proof.…”

Section: Maps Extended To the Higson Coronamentioning

confidence: 99%

Monotone-light factorizations in coarse geometry

Dydak

Weighill

2018

Topology and its Applications

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We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and light using the Higson corona. Finally, we look at some coarse properties which are preserved by coarsely light maps such as finite asymptotic dimension and exactness, and make some remarks on the situation for groups and group homomorphisms.

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Higson compactification and dimension raising

Cited by 7 publications

References 18 publications

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

Linear algebra and unification of geometries in all scales

Monotone-light factorizations in coarse geometry

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