Wild Branching Surfaces and Topological 4-manifolds

We prove that any closed connected oriented topological 4-manifold is a 4-fold topological simple branched cover of S4. As relevant intermediate steps, we extend the known branched covering representation for closed PL 4-manifolds to compact bounded and open PL 4-manifolds. Specifically, we show that any compact connected oriented PL 4-manifold with boundary is a simple branched cover of S4 minus some 4-balls, and then, by an exhaustion procedure, we obtain any open connected oriented 4-manifold as a simple branched cover of S4 minus a copy of its end space. A key point for such results is a lemma, that is interesting in its own, on the cobordism equivalence of simple branched coverings of S3, that represent the same 3-manifold and are extendable to coverings of B4 branched over ribbon surfaces.