This is a follow-up to this question, and Mark Grant's excellent answer. The answer shows that the answer is yes, but what if one were under a strange compunction to actually construct the submanifold that the surface bounds (assuming that the ambient space is given by a triangulation)? Does the spectral sequence machine imply an algorithm, and if not, is there some other way?
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asked Mar 5, 2014 at 19:26
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$\begingroup$ The spectral sequence should give you an algorithm in principle. In low dimensions like that, the obstructions are usually readily dealt with quite explicitly. A not very effective procedure to find the 3-manifold would be to take the exterior of the surface, triangulate it and look for the 3-manifold as a "normal" 3-manifold in that triangulation. One might have to subdivide the triangulation for the 3-manifold to appear as a vertex-normal solution. $\endgroup$– Ryan BudneyCommented Mar 5, 2014 at 19:38
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$\begingroup$ Good question! There should be some way to take the null-homology and resolve the singularities, and perhaps this can be made algorithmic, but I don't see how this should be related to the spectral sequence. One question: is it clear that the surface bounds a submanifold? I don't see why the 3-manifold needs to be embedded. $\endgroup$– Mark GrantCommented Mar 6, 2014 at 7:20
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$\begingroup$ @MarkGrant No, now that you mention it, it is not clear, though I have a bit set that this is true, for some reason... $\endgroup$– Igor RivinCommented Mar 6, 2014 at 12:45
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$\begingroup$ The classical method of Kneser for eliminating singularities should work in this setting. (cite: a la recherche de la topologie perdue, page 61. zbmath.org/?q=an:03962761) $\endgroup$– user126154Commented Mar 7, 2014 at 9:14
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