Abstract
We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R-3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S-3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S-3, then there is also an embedding in which X has a short meridian, that is, an essential curve in the boundary of X bounding a disk in S-3 \ X with length bounded by a computable function of the number of tetrahedra of X.