渐近维数的上同调方法
Cohomological approach to asymptotic dimension
摘要：
在有界上同调的基础上引入了渐近上同调的概念，定义了度量空间的上同调渐近维数$\as_{\Z} X$.证明了当渐近维数$\as X$为有限时，它与渐近维数一致.然后利用这一事实构造了一个渐近维数有限的有界几何度量空间$X$的例子，其中$\as(X\times\R)=\as X$。特别地，对于这个例子，由Roe粗上同调定义的粗渐近维数严格地小于其渐近维数。
英文摘要：
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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