3-球面中的可嵌入性是可判定的
Embeddability in the 3-sphere is decidable
摘要：
我们证明了以下算法问题是可判定的：给定一个$2$维单纯复形，它可以嵌入（拓扑，或等价地，分段线性）在$\mathbf{R}^3$？通过一个已知的约简，它足以决定一个给定的三角形3-流形$X$到3-球面$S^3$的嵌入性。主要步骤，这使我们能够简化$X$和递归，是在证明，如果$X$可以嵌入在$S^3$，那么也有一个嵌入，其中$X$有一个短子午线，即，$X$边界上的一条基本曲线，以$S^3\setminus X$为界，其长度由$X$四面体数目的可计算函数所限定。
英文摘要：
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 $\mathbf{R}^3$? 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 essential curve in the boundary of $X$ bounding a disk in $S^3\setminus X$ with length bounded by a computable function of the number of tetrahedra of $X$.
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