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Coarse Sheaf Cohomology

Hartmann, Elisa 1
1 Fakultät für Mathematik (MATH), Karlsruher Institut für Technologie (KIT)


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.

Verlagsausgabe §
DOI: 10.5445/IR/1000163477
Veröffentlicht am 26.10.2023
DOI: 10.3390/math11143121
Zitationen: 1
Web of Science
Zitationen: 1
Zitationen: 1
Cover der Publikation
Zugehörige Institution(en) am KIT Fakultät für Mathematik (MATH)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2023
Sprache Englisch
Identifikator ISSN: 2227-7390
KITopen-ID: 1000163477
Erschienen in Mathematics
Verlag MDPI
Band 11
Heft 14
Seiten Art.Nr.: 3121
Vorab online veröffentlicht am 14.07.2023
Schlagwörter coarse geometry, sheaf cohomology, Grothendieck site, Higson corona, Roe coarse cohomology
Nachgewiesen in Dimensions
Web of Science
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