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Cohomological approach to asymptotic dimension

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Abstract

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.

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  1. Department of Mathematics, University of Florida, P.O. Box 118105, 358 Little Hall, Gainesville, FL, 32611-8105, USA

    A. Dranishnikov

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  1. A. Dranishnikov
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Dranishnikov, A. Cohomological approach to asymptotic dimension. Geom Dedicata 141, 59–86 (2009). https://doi.org/10.1007/s10711-008-9343-0

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  • DOI: https://doi.org/10.1007/s10711-008-9343-0

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