Classifying spaces are obviously unique up to homotopy type. I am wondering, whether under stronger conditions, one can also say that they are unique up to homeomorphism. In particular, suppose $\Gamma$ is a group and there exist a model $X$ for $B\Gamma$, which is closed (compact without boundary). Suppose $Y$ is also a model for $B\Gamma$ and $Y$ is also closed. In my baby examples it seems reasonable that $X\cong Y$. Is this always true?
Furthermore, if $X$ and $Y$ are models for $B\Gamma$ and $X$ is a closed $n$-dimensional manifold and $Y$ is also an $n$-dimensional manifold. Is it true that $Y$ is closed as well?