The purpose of this article is to define and study new invariants of topological spaces: the $p$‐adic Betti numbers and the $p$‐adic torsion. These invariants take values in the $p$‐adic numbers and are constructed from a virtual pro‐ $p$ completion of the fundamental group. The key result of the article is an approximation theorem which shows that the $p$‐adic invariants are limits of their classical analogues. This is reminiscent of Lück's approximation theorem for $L^2$‐Betti numbers.

中文翻译：

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关于拓扑不变量的p-adic极限

本文的目的是定义和研究拓扑空间的新不变量： $p$-adic贝蒂数和 $p$adic扭转。这些不变量取值$p$-adic数，由虚拟pro-构成 $p$基本小组的完成。本文的主要结果是一个近似定理，表明$p$-adic不变量是其经典类似物的局限性。这让人联想到吕克（Lück）的近似定理${\mathrm{大号}}^2$‐贝蒂数字。