Cohomological approach to asymptotic dimension

Details Cohomological approach to asymptotic dimension Cohomological approach to asymptotic dimension

2006-08-09

arxiv.org/abs/math/0608215v1

Mathematics

Metric Geometry

30 pages

Scientific paper

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ 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 $\as(X\times\R)=\as X$. 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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