Coarse Homotopy on metric Spaces and their Corona

Hartmann, Elisa

doi:10.48550/arxiv.1907.03510

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…A section in this paper transfers sheaves on a proper metric space to sheaves on its Higson corona. For this purpose we give a definition of the Higson corona which is equivalent to the usual one [Har19c].…”

Section: Coarse Cohomology By Roe and The Higson Corona Elisa Hartmannmentioning

confidence: 99%

Coarse sheaf cohomology

Hartmann¹

2022

Preprint

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A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0 they see the number of ends of the space. In this paper a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus we can use topological tools on compact Hausdorff spaces in our computations. In particular if the asymptotic dimension of a proper metric space is finite then higher cohomology groups vanish. We compute a few examples. As it turns out finite abelian groups are best suited as coefficients on finitely generated groups.

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Section: Coarse Cohomology By Roe and The Higson Corona Elisa Hartmannmentioning

confidence: 99%

Coarse sheaf cohomology

Hartmann¹

2022

Preprint

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“…We now describe the results on the specific examples in detail. In particular the boundary of the Higson compactification retains information about the coarse structure since the Higson corona is a faithful functor [Har19b]. This way not much information is lost if we restrict our attention to the boundary of a coarse compactification when studying coarse metric spaces.…”

Section: Introductionmentioning

confidence: 99%

Coarse compactifications of proper metric spaces

Hartmann¹

2020

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This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on X that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of X which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity.Based on this foundational work we study examples for coarse compactifications Higson compactification, Freudenthal compactification and Gromov compactification. For each example we characterize the bounded functions which can be extended to the coarse compactification and the corresponding large-scale proximity relation.We provide an alternative proof for the property that the Higson compactification is universal among coarse compactifications. Furthermore the Freudenthal compactification is universal among coarse compactifications with totally disconnected boundary. If X is hyperbolic geodesic proper then there is a closed embedding ν(R+) × ∂X → ν(X). Its image is a retract of ν(X) if X is a tree. Contents 0 Introduction 1 Notions in coarse geometry 2 The original definition 3 Large-scale proximity relations 4

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Coarse Homotopy on metric Spaces and their Corona

Cited by 2 publications

References 14 publications

Coarse sheaf cohomology

Coarse sheaf cohomology

Coarse compactifications of proper metric spaces

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