Cohomological dimension theory in coarse geometry

当我们从渐近维度理论的观点开始研究粗几何时，我们面临着一个与嵌入问题相关的重要问题。也就是说，我们研究了什么样的n维紧致度量空间可以嵌入到n个一维紧致度量空间的乘积中。首先，利用几何结构和一维上同调群的秩来确定一类n维拓扑广义流形，它可以嵌入到n维紧致度量空间的乘积中。其次，我们将该方法应用于任意n维紧度量空间，利用n维Cech上同调群的平凡性和1维Cech上同调群的秩给出了判定可嵌入性的准则。

As we started to investigate coarse geometry from the viewpoint of asymptotic dimension theory, we faced an important problem related to embedding problems. Namely, we investigated the problem what kind of n-dimensional compact metric spaces can be embedded into a product of n one-dimensional compact metric spaces. First, we tried to determine a class of n-dimensional topological and generalized manifolds which can be embedded into a product of n one-dimensional compact metric spaces by using geometric structures and the rank of 1-dimensional cohomology groups. Next, we applied this method to arbitrary n-dimensional compact metric spaces to determine embeddability and succeeded to give a criterion by using the triviality of n-dimensional Cech cohomology groups and the rank of 1-dimensional Cech cohomology groups.