Twisted coefficients on coarse spaces and their corona
In recent years, the author of this paper has developed a theory of twisted coarse cohomology for metric spaces [Math. Slovaca 70, No. 6, 1413--1444 (2020; Zbl 1505.51003); ``Coarse sheaf cohomology'', Preprint, \url{arXiv:2205.01635}]. As far as I can see, this theory utilizes a notion of \textit{coarse covers} and considers sheaf cohomology of induced Grothendieck topologies. Moreover, it can be computed from the Higson corona of the space.\N\NIt is my hypothesis that the genesis of the paper under review can be traced to the author's discovery of the works by \textit{Sh. Kalantari} and \textit{B. Honari} [Rocky Mt. J. Math. 46, No. 4, 1231--1262 (2016; Zbl 1355.53019)] and \textit{I. V. Protasov} [Topology Appl. 149, No. 1--3, 149--160 (2005; Zbl 1068.54036)], \textit{I. V. Protasov} and \textit{S. V. Slobodianiuk} [Ukr. Math. J. 67, No. 12, 1922--1931 (2016; Zbl 1388.54021)]. These authors found methods to describe Higson coronas by associating a \textit{proximity structure} with any metric space (or, more generally, ay ballean space) and considering an associated compactification.\NConsequently, it must have been natural for the author to attempt to reconcile her approach to coarse cohomology via coarse covers with the language of proximities, (coarse) ultrafilters, and Smirnov compactifications presented in [Kalantari and Honari, loc. cit.; Protasov, loc. cit.; Protasov and Slobodianiuk, loc. cit.]. These efforts have proven successful and the paper under review is the result.\N\NTo the best of my understanding, the main results can be summarized as follows: for \textit{coarsely proper} (i.e. bounded geometry) metric spaces, the coarse sheaf cohomology of \(X\) coincides with the sheaf cohomology of its Higson corona \(\nu(X)\). Moreover, the homomorphisms that a \textit{coarse map} \(f\colon X\to Y\) induces at the level of these cohomology groups agree as well.\N\NA note on nomenclature: there are various sets of conventions in the field of coarse geometry.\NFor clarity, I should mention that in this paper \textit{coarse maps} are defined in the sense of Roe (i.e., a function that is controlled and proper). A \textit{proper coarse space} is a metric space that is coarsely equivalent to a locally finite metric space (a property also known as \textit{bounded geometry}).\N\NThis paper is not self-contained, most likely because of its origin. Readers interested in twisted coarse cohomology would need to consult the author's previous works, and the constructions of coarse coronas are likely better understood through the other sources provided. The exposition of the material in this manuscript would have benefited from additional refinement: as it currently stands, it is not always easy to follow.
scientific article; zbMATH DE number 7899637
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