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, and it admits silly answers like this one.

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 the problem immediately disappears because any such set is non-recursive by Rice's theorem.

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.)