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My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?

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Are you asking about 3-dimensional CW complexes? If not, could you let us know what you mean by a "3-dimensional complex". – Ryan Budney Oct 4 '13 at 18:39
up vote 9 down vote accepted

No, in higher dimensions there is no analogue of Kuratowski's theorem. See thesis of Anna Gundert for a leisurely overview, and the references. There is also a recent work of Matoušek, Tancer and Wagner that addresses how to determine if a complex is embeddable. They have a nice table for different dimensions on page 4 from it transpires it is not known how hard the problem is.

Addition on 21 Jul 2014: The problem is now known to be decidable. See the recent paper by Matoušek, Sedgwick, Tancer, and Wagner (you want Corollary 1.2).

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