# Branched Coverings of the $4$-sphere branched along a knotted surface

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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Look at the first Greg's comment here mathoverflow.net/questions/8697/ramified-cover-of-4-sphere/… –  Anton Petrunin Mar 10 '12 at 16:55
@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. –  Scott Carter Mar 11 '12 at 0:46
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." –  Anton Petrunin Mar 11 '12 at 1:55
OK, Now I understand. You are empowered to close the question, I think so please do. –  Scott Carter Mar 11 '12 at 2:41