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It is shown that the \(\ell_2\)-Betti numbers of an amenable covering of a finite cell-complex can be approximated by ordinary Betti numbers of the finite Fœlner subcomplexes. This is a new proof, using simple homological arguments, of a recent result of Dodziuk and Mathai.

Approximating $\ell_2$ -Betti numbers of an amenable covering by ordinary Betti numbers