Is it true that any closed oriented $4$-dimensional manifold can be obtained as a result of the following construction: 


Take $S^4$ with a finite collection of immersed closed 2-manifolds (with transversal intersections and self-intersections) and construct ramified cover of $S^4$ with a ramification of order at most 2 only at these submanifolds.


**Comments:**


* Two related questions: [Ramified covers of 3-torus](https://mathoverflow.net/questions/5546/ramified-covers-of-3-torus), [Ramified covers of $S^n$](https://mathoverflow.net/questions/5618/ramified-covers-of-sn)
* According to Feighn's [Branched covers according to J.W. Alexander](http://www.raco.cat/index.php/CollectaneaMathematica/article/view/56965) any closed oriented 4-manifold is a branched cover of $S^4$ with a ramification along 2-skeleton of 4-tetrahedron embedded in $S^4$ (which is not at all a 2-manifold).



Ramified cover of 4-sphere
The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as $4$-fold branched covers of $S^4$. Topology 34 (1995), no. 3, 497--508. Any closed, orientable PL 4-manifold can be expressed as a 4-fold simple branched covering of S4 branched along an immersed surface with only transverse double points. It is apparently still an open question whether the branch set can be chosen to be nonsingular. A simple branched covering of degree d is a branched covering in which each branch point is covered by d-1 points, only one of which is singular, of local degree 2.


