that $$\liminf_{n \to \infty} \frac{\dim_{k}\bigl(H_i(G_n\backslash X;k)\bigr)}{[G:G_n]} 
\ge 
\lim_{n \to \infty} \frac{\dim_{k}\bigl(H_i(G_n\backslash X;{{\mathbb Q}})\bigr)}{[G:G_n]} 
= b_i^{(2)}(X;{{\mathcal N}}(G)),$$ where the latter term denotes the $i$-th $L^2$-Betti number of $X$. In particular we get from Theorem \[the:dim\_approximation\_over\_fields\] for a torsionfree amenable group $G$ with no zero-divisors in $kG$ that $$\begin{aligned}
\dim^{{\operatorname{Ore}}}_{kG}\bigl(H_i(X;k)\bigr) 
 = 
\lim_{n \to \infty} \frac{\dim_{k}(H_i(G_n\backslash X;k))}{[G:G_n]} 
& \ge 
\lim_{n \to \infty} \frac{\dim_{k}(H_i(G_n\backslash X;{{\mathbb Q}}))}{[G:G_n]} 
\\
& = 
b_i^{(2)}(X;{{\mathcal N}}(G))
\\
& = 
\dim^{{\operatorname{Ore}}}_{{{\mathbb C}}G}\bigl(H_i(X;{{\mathbb C}})\bigr).\end{aligned}$$ This inequality is in general not an equality as the next example shows.

\[exa:S1\_wedge\_Sd\_cupDd\_plus\_1\] Fix an integer $d \ge 2$ and a prime number $p$. Let $f_p \colon S^d \to S^d$ be a map of degree $p$ and denote by $i \colon S^d \to S^1 \vee S^d$ the obvious inclusion. Let $X$ be the finite $CW$-complex obtained from $S^1 \vee
S^d$ by attaching a $(d+1)$-cell with attaching map $i \circ f^d \colon S^d
\to S^1 \vee S^d$. Then $\pi_1(X) = {{\mathbb Z}}$. Let $\widetilde{X}$ be the universal covering of $X$ which is a finite free ${{\mathbb Z}}$-$CW$-complex. Denote by $X_n$ the covering of $X$ associated to $n \cdot {{\mathbb Z}}\subseteq {{\mathbb Z}}$. The cellular ${{\mathbb Z}}C$-chain complex of $\widetilde{X}$ is concentrated in dimension $(d+1)$, $d$ and $1$ 