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Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be triangularized and hence be described by a finite amount of combinatorial data.

In higher dimensions this argument doesn't work anymore. So it still true for $n\ge 4$ that the set of homeomorphism classes of compact, connected topological $n$-manifolds (without boundary) is countable?

(I'd be also interested in the same question for diffeomorphism classes of compact smooth manifolds.)

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    $\begingroup$ For diffeomorphism classes the answer is in the affirmative, as any smooth manifold can be triangulated and within each $PL$-class there are finitely many diffeomorphism classes. $\endgroup$ – Alex Degtyarev Feb 21 '15 at 11:53
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    $\begingroup$ You also want to assume connectedness, since otherwise any disjoint union of closed manifolds would again be a closed manifold, and these disjoint unions can be arbitrarily big. $\endgroup$ – Tobias Fritz Feb 21 '15 at 13:18
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    $\begingroup$ @TobiasFritz They would be finite disjoint unions. $\endgroup$ – Alex Degtyarev Feb 21 '15 at 13:35
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    $\begingroup$ @Alex Degtyarev: can you give references? $\endgroup$ – ACL Feb 21 '15 at 14:55
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    $\begingroup$ @Wlodzimierz: what do you mean exactly? $\endgroup$ – ACL Feb 21 '15 at 14:57
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It was shown in

J. Cheeger and J. M. Kister, Counting topological manifolds. Topology 9, 1970 149–151.

that there are only countably many compact manifolds up to homeomorphism (even allowing boundaries).

Here is a link to the article.

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    $\begingroup$ You may google for "homotopy Holsztyński Kister Mather. When I do it then the first reference is to MO (of course). That MO answer provides three references to papers of which the first two are pointed to by: J. Cheeger and J. M. Kister, Counting topological manifolds. Topology 9, 1970 149–151, just mentioned by you. At the time Jim Kister knew very well about my result (and some my related) but didn't tell me about the join paper by him and Cheeger--most like my result preceded theirs but mine took much longer to get it in print. $\endgroup$ – Włodzimierz Holsztyński Feb 21 '15 at 20:32
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    $\begingroup$ The join result by Cheeger and Kister (1970) is much-much sharper than the other three papers by Mather(1965), Kister(1968), Holsztyński (1971). The last one is the most general (and perhaps the simplest too) among the homotopical results. $\endgroup$ – Włodzimierz Holsztyński Feb 21 '15 at 20:42
  • $\begingroup$ If one assumes some quantitative topology, then one indeed also gets quantitative bounds on the number of homotopy type or homeomorphism type. This is the geometric finiteness theorem obtained by Grove-Peterson-Wu 1990. $\endgroup$ – Changyu Guo Feb 22 '15 at 15:21
  • $\begingroup$ A link to the MO answer of Włodzimierz Holsztyński : mathoverflow.net/a/137649 $\endgroup$ – j.c. Oct 23 '15 at 16:58

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