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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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Re: the first sentence, that depends on how you define classifying spaces. If you define them in terms of the functor they represent then of course they are unique up to homotopy type. But if you define them as quotients of weakly contractible spaces by free actions then this seems less clear to me (but then again I have not really studied this). –  Qiaochu Yuan Jun 21 '12 at 6:58
    
Thank you. I only want to consider discrete groups. –  Earthliŋ Jun 21 '12 at 7:00
    
But even then I do not think it is obvious that Eilenberg-MacLane spaces are unique up to homotopy type (again unless one defines them via the functor they represent). This was first proven by Hopf, I think. –  Qiaochu Yuan Jun 21 '12 at 7:01
    
Hm, I see. Do you have a reference for this? –  Earthliŋ Jun 21 '12 at 7:04
    
For CW-complexes this is Theorem 1B.8 in Hatcher. –  Qiaochu Yuan Jun 21 '12 at 7:07

2 Answers 2

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! –  Earthliŋ Jun 21 '12 at 7:01

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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Mostow rigidity proves the conjecture for $n \ge 3$. For $n = 2$ we just look at the classification of compact surfaces, I guess. –  Qiaochu Yuan Jun 22 '12 at 14:16
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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. –  mland Jun 23 '12 at 9:36

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