Earlier this year it was asked on MO, "http://mathoverflow.net/questions/198098/are-there-only-countably-many-compact-topological-manifolds"  Thanks to Cheeger and Kister, the answer is yes.  On the other hand, Manolescu recently debunked the triangulation conjecture.  A natural follow-up question asks if there is some other way to enumerate topological n-manifolds, in the sense of creating a Turing machine that will eventually output an example from every homeomorphism class of topological manifolds, given enough time.

Of course, for $n \leq 3$, TOP = PL, so I'm really interested in the cases $n\geq 4$.  It's entirely possible that the answer still depends on $n$, so you can interpret the question with either $n$ fixed or variable.

If the answer is no, is it known how hard the problem of enumerating manifolds is?  Is it harder than the halting problem?  More precisely, is the Turing degree known?