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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?

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    Anyway, a positive result: if I'm not horribly mistaken this follows for compact hyperbolic $n$-manifolds, $n \ge 3$ by Mostow rigidity (http://en.wikipedia.org/wiki/Mostow_rigidity_theorem).2012-06-21

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Regarding your last question:

If $X$ and $Y$ are homotopic $n$-dimensional manifolds and one is closed, so is the other. (They are homotopic by assumption, thus have isomorphic homology. Since one can detect using $H^n$ whether or not an $n$-manifold is closed, they are either both closed or neither is.)

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    Brilliant. Thank you!2012-06-21
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There is something called an aspherical manifold. This is a closed manifold whose universal cover is contractible. In particular any aspherical manifold $M$ is a model for $B\pi$ where $\pi=\pi_1(M)$. There are many examples of aspherical manifold, for example any closed manifold of negative sectional curvature (e.g. hyperbolic manifolds) are aspherical by the Theorem of Cartan-Hadamard, that the exponential map is a then covering map.

Now there is a beautiful conjecture due to Borel (the Borel Conjecture) which states that any two aspherical manifolds $M$ and $N$ with isomorphic fundamental group $\pi$ are homeomorphic. Even more the conjecture predicts that any homotopy equivalence $f: M \to N$ is homotopic to a homeomorphism.

Recently there has been a lot of work concerning this conjecture due to a stronger conjecture, the Farrell-Jones Conjecture. This is a conjecture about algebraic $K$ and $L$ theory of group rings. The Farrell-Jones Conjecture for $K$ and $L$ theory together imply the Borel Conjecture. Moreover the Farrell-Jones Conjecture has been proven for a quite large class of groups including hyperbolic groups, $CAT(0)$-groups and many more.

A lot of work on the Farrell-Jones Conjecture is due to Wolfgang Lück ( professor at Bonn university ) and you might want to look at some of his survey articles concerning these kinds of questions. You can find them on his homepage, http://www.math.uni-bonn.de/ag/topo/members for a link to that.

Moreover I should mention that there is a theorem called "Mostow rigidity" which proves the Borel conjecture for hyperbolic manifolds, and this is much older than the work on Farrell-Jones Conjecture.

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    yes. Sorry for forgetting about the dimension. In fact in dimension 2 one needs more than the classification of compact surfaces because the Borel conjecture says that even every homotopy equivalence is homotopic to a homeomorphism, which does not follow from the classification. Yet there is a theorem saying that $\pi_0(Homotopy equivalences) \to \pi_0(homeomorphisms)$ induced by the inclusions is a bijection for a compact surface.2012-06-23