作为$S^2$的对称立方体的$\CC P^3$的三角剖分
A triangulation of $\CC P^3$ as symmetric cube of $S^2$
摘要：
对称群$S_3$通过坐标置换作用在$S^2 \times S^2 \times S^2$上，商空间$(S^2 \times S^2 \times S^2)/S_3$同胚于复射影空间$\CC P^3$。本文对$S^2 \times S^2 \times S^2$的64顶点标准胞元$S^2_4 \times S^2_4 \times S^2_4$构造了一个124顶点的单纯细分$(S^2 \times S^2 \times S^2)_{124}$，使得$S_3$ -作用自然地扩展到$(S^2 \times S^2 \times S^2)_{124}$上.此外，$(S^2 \times S^2 \times S^2)_{124}$上的$S_3$ -动作是“好的”，使得商单纯复形$(S^2 \times S^2 \times S^2)_{124}/S_3$是$\CC P^3$的30顶点三角剖分$\CC P^3_{30}$。换句话说，我们构造了分支覆盖$S^2 \times S^2 \times S^2 \to \CC P^3$的单纯实现$(S^2 \times S^2 \times S^2)_{124} \to \CC P^3_{30}$。最后，我们将Lutz的BISTELLAR程序应用于$\CC P^3_{30}$，得到了$\CC P^3$的18顶点2邻域三角剖分$\CC P^3_{18}$。$\CC P^3_{18}$的自同构群是平凡的。可以回想起，根据Arnoux和Marin的结果，$\CC P^3$的任何三角测量都需要至少17个顶点。因此，$\CC P^3_{18}$接近顶点最小，如果不是实际上顶点最小的话。此外，到目前为止，还没有明确的$\CC P^3$三角测量。
英文摘要：
The symmetric group $S_3$ acts on $S^2 \times S^2 \times S^2$ by coordinate permutation, and the quotient space $(S^2 \times S^2 \times S^2)/S_3$ is homeomorphic to the complex projective space $\CC P^3$. In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \times S^2 \times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \times S^2_4 \times S^2_4$ of $S^2 \times S^2 \times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \times S^2 \times S^2)_{124}$. Further, the $S_3$-action on $(S^2 \times S^2 \times S^2)_{124}$ is "good", so that the quotient simplicial complex $(S^2 \times S^2 \times S^2)_{124}/S_3$ is a 30-vertex triangulation $\CC P^3_{30}$ of $\CC P^3$. In other words, we construct a simplicial realization $(S^2 \times S^2 \times S^2)_{124} \to \CC P^3_{30}$ of the branched covering $S^2 \times S^2 \times S^2 \to \CC P^3$. Finally, we apply the BISTELLAR program of Lutz on $\CC P^3_{30}$, resulting in an 18-vertex 2-neighbourly triangulation $\CC P^3_{18}$ of $\CC P^3$. The automorphism group of $\CC P^3_{18}$ is trivial. It may be recalled that, by a result of Arnoux and Marin, any triangulation of $\CC P^3$ requires at least 17 vertices. So, $\CC P^3_{18}$ is close to vertex-minimal, if not actually vertex-minimal. Moreover, no explicit triangulation of $\CC P^3$ was known so far.
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