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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, i.e., an essentialarXiv:1402.0815v2 fatcat:b63l67yc3zhodolvctcm7dhmry