A pullback diagram in the coarse category

Hartmann, Elisa

doi:10.48550/arxiv.1907.02961

Cited by 2 publications

(4 citation statements)

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“…There are several notions of homotopy on the coarse category which are all equivalent in some way. The homotopy theory we are going to employ uses the asymptotic product as coarse substitute for a product and the first quadrant in R 2 equipped with the Manhattan metric as a coarse substitute for an interval [Har19b]. In effect this homotopy theory and the other coarse homotopy theories are only of use if one wants to compute cohomology of R n and maybe Riemannian manifolds.…”

Section: ȟQ Ct (T ; A) =mentioning

confidence: 99%

“…The paper [Har19b] shows that X * I is the pullback of d(•, x 0 ) and d(•, (0, 0)). Moreover we can define a well-defined homotopy theory: If X is a metric space define maps…”

Section: Thus (Cl(umentioning

confidence: 99%

“…The paper [Har19b] shows coarse homotopy is an equivalence relation and compares this theory with other homotopy theories on the coarse category.…”

Section: Thus (Cl(umentioning

confidence: 99%

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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: ȟQ Ct (T ; A) =mentioning

confidence: 99%

“…The paper [Har19b] shows that X * I is the pullback of d(•, x 0 ) and d(•, (0, 0)). Moreover we can define a well-defined homotopy theory: If X is a metric space define maps…”

Section: Thus (Cl(umentioning

confidence: 99%

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Coarse sheaf cohomology

Hartmann¹

2022

Preprint

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“…The asymptotic product of two metric spaces has been introduced in [10] as the limit of a pullback diagram in the coarse category. Note [11, Theorem 1] shows the following: If X, Y are hyperbolic coarsely proper coarsely geodesic metric spaces then X * Y is hyperbolic coarsely proper coarsely geodesic and therefore its Gromov boundary ∂(X * Y ) is defined.…”

Section: Introductionmentioning

confidence: 99%