On $p$-adic limits of topological invariants

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 L\"uck's approximation theorem for $L^2$-Betti numbers. After an investigation of basic properties and examples we discuss the $p$-adic analog of the Atiyah conjecture: When do the $p$-adic Betti numbers take integer values? We establish this property for a class of spaces and discuss applications to cohomology growth.