On branched covering representation of 4-manifolds

Assuming M to be a connected oriented PL 4-manifold, our main results are the following: (1) if M is compact with (possibly empty) boundary, there exists a simple branched covering p : M → S4 − Int(B41 ∪ · · · ∪ B4n), where the PL 4-balls B4i) are pairwise disjoint, n ≥ 0 is the number of boundary components of M; (2) if M is open, there exists a simple branched covering p : M → S4 − End M, where End M is the end space of M tamely embedded in S4. In both cases, the degree d(p) and the branching set Bp of p can be assumed to satisfy one of these conditions: (1) d(p) = 4 and Bp is a properly self-transversally immersed locally flat PL surface; (2) d(p) = 5 and Bp is a properly embedded locally flat PL surface. In the compact (respectively, open) case, by relaxing the assumption on the degree we can have B4 (respectively, R4) as the base of the covering. A crucial technical tool used in all the proofs is a quite delicate cobordism lemma for coverings of S3, which also allows us to obtain a relative version of the branched covering representation of bounded 4-manifolds, where the restriction to the boundary is a given branched covering. We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented topological 4-manifold is a 4-fold branched covering of S3. According to almost smoothability of 4-manifolds, this branched covering could be wild at a single point.