Spaces that are finitely covered by manifolds Suppose $X \to Y$ is a finite-sheeted covering of CW-complexes. Moreover, assume that the total space is homotopy equivalent to a (closed, connected, smooth) manifold $M$. I am interested in conditions to impose on the map $X \to Y$ so that we can find a manifold $N \simeq Y$ and a covering $M \to N$ making the diagram $$ \begin{array}{ccc}
    X & \simeq & M \\
    \downarrow && \downarrow \\
    Y & \simeq & N
   \end{array}
$$
commute. In general, this is not possible, for example we could take any finite group $G$ and $$X = M \times EG \to M \times BG = Y.$$ Are there counterexamples for $Y$ a finite CW-complex? On the other side, are there conditions to impose so that we actually can deduce the existence of $N$?
 A: Conditions where $N$ exists are for example when $M$ is a non-positively curved locally symmetric manifold, by a theorem of Mostow (for hyperbolic manifolds) and Margulis in general. Mostow's theorem is generalized to manifolds with hyperbolic fundamental group whose Gromov boundary is a sphere of dimension at least 5.   So if one started with such a manifold, any group containing the fundamental group with finite index and torsion-free would have to be a manifold group. 
More generally, it is conjectured that finitely presented groups which are $PD(n)$ groups are the fundamental groups of closed $n$-manifolds (Conjecture 3.4). If true, this would show that $N$ exists when $X$ is aspherical. This conjecture would follow for $n\geq 5$ from the Novikov conjecture. Many cases of this conjecture are known (for groups satisfying certain restrictions). See also Sections 5 and 8 of this paper. 
A: Indeed, C. Thomas, 3-Manifolds and PD(3)-groups, in "Novikov conjectures", Vol. 2, gives a reference to Swan's example. Namely, there is a finite complex $X$ which is a homotopy  3-sphere, a free $S_3$-action $S_3\times X\to X$, which cannot be realized (up to homotopy) as a free $S_3$-action on $S^3$, since $S_3$ cannot act freely on $S^3$ (the 3-sphere). 
