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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.


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up vote 8 down vote accepted

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.

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Actually in 2002 Iori and Piergallini got rid of the double points too. The branch set can be non-singular with 5 sheets. What is still open is 4 sheets and non-singular branch locus. (Google is our friend.) – Greg Kuperberg Dec 12 '09 at 22:44
Thank you so much :) – Anton Petrunin Dec 12 '09 at 23:40
Thanks a lot for the answer! At the same time, since the results of these articles are formulated in terms of PL manifolds and locally flat PL surfaces I would like to know something additional. As far as I understand PL manifolds of dimension 4 have canonical smooth structures. Is it true that locally flat PL surfaces in a 4-dim PL manifold are isotopic to smooth surface for this smooth strucutre? In over words, are these results still hold in smooth cathegory? – Dmitri Dec 12 '09 at 23:43
Yes, that part works easily. You just check that the link of each vertex of the surface (and more trivially the link of each edge) is something that allows the vertex to be smoothed. The locally flat condition says that the link of each vertex is an unknot in its 3-sphere. – Greg Kuperberg Dec 13 '09 at 0:02

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