We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded in R3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X in the 3-sphere. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded, then there is also an embedding in which X has a short meridian, i.e., an essential curve in the boundary of X bounding a disk in S3∖X, whose length is bounded by a computable function of the number of tetrahedral of X.
This is joint work with Matoušek, Tancer and Wagner.

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