This paper aims to use Cartan’s original method in proving Theorems A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement’s argument to show the topological dimension of a paracompact topological manifold is less than or equal to its real dimension.

中文翻译：

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嘉当方法及其在层上同调中的应用

本文旨在使用嘉当在闭立方体上证明定理 A 和 B 的原始方法，为闭立方体上的束上同调消失提供不同的证明，如果 (i) 次数超过其真实维数或 (ii) 束为（局部）恒定且次数为正。在第一种情况下，我们可以进一步利用 Godement 的论证来证明仿紧拓扑流形的拓扑维数小于或等于其真实维数。