Are compact topological $n$-manifolds recursively enumerable? Earlier this year it was asked on MO, "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?
Edit in response to comments below: I do not mean to jump the gun. To even have a hope that the answer to the question is yes, one would have to have some finitely computable description of topological manifolds.  As BjørnKjos-Hanssen indicates in comments, this might take the form of some sequence of approximations.   If a direct answer to my question seems out of reach, I would be happy with an answer explaining what is and isn't known.  (I also removed the madness about reference to Turing degrees above.)
 A: In a note of Freedman and Zuddas, they show that this is true for dimensions $\geq 4$.
In the "Background" section of the paper, they describe the solution in the higher dimensional case using surgery theory, but without any references.
Then they proceed to describe the 4-dimensional case. Here they use the fact that the complement of a point in a 4-manifold is smoothable. Hence one can describe a triangulation of a finite part of the complement of a point, together with a certificate of a 3-sphere tamely embedded.
It's frustrating that they don't give any references for the higher-dimensional case, but since you're in Santa Barbara, you could probably saunter over to Station Q to get the details from Mike once campus opens again.
A: At the risk of getting on everyone's nerves (and special apologies to the OP), I would still maintain that the question in this form is too vague.
If I understand the suggestions in the comments correctly, they amount to something like this: interpret the output of a Turing machine as a sequence of points (possibly empty or finite) $x_0,x_1,x_2,\ldots\in\mathbb Q^{2n+1}$. Now we can ask: for what $e$ will TM number $e$ output a sequence of points whose closure in $\mathbb R^{2n+1}$ is an (embedded) manifold of the desired type? (This is certainly not exactly what Bjorn suggested, but it feels close enough.)
However, if we formalize like this, then part of the problem immediately disappears because any such set is non-recursive by Rice's theorem and this doesn't feel very satisfying because it had nothing to do with manifolds. (Admittedly, we could still ask about other properties of my set of $e$'s.)
