$k$ is a field. Let $X$ be a connected pointed CW-complex such that the homology 
$H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$.  Suppose that we have continuous mapping 
$$r: X\rightarrow K(\pi_{1}(X),1) $$
from the space $X$ to the Eilenberg-MacLane space $K(\pi_{1}(X),1)$ inducing an isomorphism on fundamental group $\pi_{1}$. Suppose also that there is a mapping 
$$ i: K(\pi_{1}(X),1) \rightarrow X $$
such that:

 1. $r\circ i= id$
 2. the homotopy cofiber of $i$ is homotopy equivalent to a finite $CW$-complex.
 3. and $H_{n}(r;k): H_{n}(X;k)\rightarrow H_{n}(K(\pi_{1}(X),1);k)$ is an isomorphism for any $n\in \mathbb{N}$.

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> My question is the following: Is the homology of $\tilde{X}$ (the
> universal covering of $X$) finite dimensional ? i.e. $H_{n}(\tilde{X};k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$? 

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