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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  →   S  4   −   Int    (   B  1  4   ∪  ⋯  ∪   B  n  4   )    , where the PL 4‐balls    B  i  4    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  →   S  4   −   End   M   , whereEnd   M    is the end space of   M   tamely embedded in    S  4   .  In both cases, the degree    d  (  p  )    and the branching set    B  p    of   p   can be assumed to satisfy one of these conditions: (1)    d  (  p  )    =    4    and    B  p    is a properly self‐transversally immersed locally flat PL surface; (2)    d  (  p  )    =    5    and    B  p    is a properly embedded locally flat PL surface. In the compact (respectively, open) case, by relaxing the assumption on the degree we can have    B  4    (respectively,    R  4   ) as the base of the covering.  A crucial technical tool used in all the proofs is a quite delicate cobordism lemma for coverings of    S  3   , 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    S  4   . According to almost smoothability of 4‐manifolds, this branched covering could be wild at a single point.

On branched covering representation of 4‐manifolds