This question has been inspired by [covering 3-torus post](https://mathoverflow.net/questions/5546/ramified-covers-of-3-torus).



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> Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away from codimension 2?
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Ramified covers of S^n
Yes. See Feighn's short [note](http://www.raco.cat/index.php/CollectaneaMathematica/article/view/56965) "Branched covers according to J.W. Alexander". 


