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?



3-complexes not embeddable in 3-space
No, in higher dimensions there is no analogue of Kuratowski's theorem. See [thesis of Anna Gundert](http://www.inf.ethz.ch/personal/gunderta/files/Diplomarbeit.pdf) for a leisurely overview, and the references. There is also a [recent work of Matoušek, Tancer and Wagner](http://arxiv.org/abs/0807.0336) 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](http://arxiv.org/abs/1402.0815) (you want Corollary 1.2).


