We prove that the problem of deciding whether a two- or three-dimensional simplicial complex embeds into R 3 is NP -hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an S 3 filling is NP -hard. The former stands in contrast with the lower-dimensional cases, which can be solved in linear time, and the latter with a variety of computational problems in 3-manifold topology, for example, unknot or 3-sphere recognition, which are in NP ∩ co- NP. (Membership of the latter problem in co-NP assumes the Generalized Riemann Hypotheses.) Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings of manifolds with boundary tori.

中文翻译：

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R 3 中的可嵌入性是 NP 难的

我们证明了决定二维或三维单纯复形是否嵌入到 R 中的问题3是NP-难的。我们的构造还表明，决定具有边界环面的 3 流形是否承认 S3填充是NP-难的。前者与低维情况形成对比，后者可以在线性时间内解决，而后者则与 3 流形拓扑中的各种计算问题形成对比，例如，在 NP ∩ 中的 unknot 或 3-sphere 识别共同NP。（co-NP 中后一个问题的成员假定广义黎曼假设。）我们的归约将可满足性实例编码为具有边界环面的 3 流形的可嵌入性问题，并广泛依赖于低维拓扑技术，最重要的是 Dehn具有边界环面的流形填充物。