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This question is related to, but apparently not exactly the same as, Ramified cover of the $4$-sphere. Piergallini, et al. have singular points on their branch loci.

Which closed orientable $4$-dimensional manifolds can be realized as simple branched coverings of $S^4$ branched along a knotted surface? By knotted surface, I mean a smooth embedding of a closed orientable (but not necessarily connected) $2$-dimensional manifold. By simple I mean if the covering has $n$-sheets, then along the knotted surface the cover has $n-1$ sheets. Thus on the singular sheet the covering is $2$-to-$1$, as $z\mapsto z^2$.

I am especially interested in the case when $n=2$ or $3$ in which I can give an explicit immersion thereof into $S^4\times D^2 \subset of S^6$ in which the projection is the covering map.

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  • $\begingroup$ Look at the first Greg's comment here mathoverflow.net/questions/8697/ramified-cover-of-4-sphere/… $\endgroup$
    – Anton Petrunin
    Commented Mar 10, 2012 at 16:55
  • $\begingroup$ @Anton, Thanks, the way I am reading their main statement is that the branching set is a locally flat PL surface. It is conceivable to me that the surface is immersed in S^4 rather than embedded. I might be missing something obvious. I'll think about this some more. $\endgroup$
    – Scott Carter
    Commented Mar 11, 2012 at 0:46
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    $\begingroup$ On the second page here arxiv.org/pdf/math/0203087v2.pdf they say $$ $$ "In the next section we show how elimination of nodes can be performed up to cobordism of coverings, after the original 4–fold covering has been stabilized by adding a fifth trivial sheet. This proves the following representation theorem." $\endgroup$
    – Anton Petrunin
    Commented Mar 11, 2012 at 1:55
  • $\begingroup$ OK, Now I understand. You are empowered to close the question, I think so please do. $\endgroup$
    – Scott Carter
    Commented Mar 11, 2012 at 2:41

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Anton's comments above answer the question.

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