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, Ramified covers of $S^n$
According to Feighn's Branched covers according to J.W. Alexander 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).
