从二十面体到$\CC P^2$和$S^2 \times S^2$的自然三角剖分
From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$
摘要：
本文给出了两种构造方法：（a）复射影平面$\CC P^{2}$的10顶点三角剖分$\CC P^{2}_{10}$作为标准球面并的子复形（$S^{2}_4$）和标准实射影平面（$\RR P^{2}_{6}$，十面体），其自同构群为$A_4$ ;（b）$S^{2} \times S^{2}$的12顶点三角剖分$(S^{2} \times S^{2})_{12}$，自同构群$2S_5$，对称群$S_5$的Schur双覆盖。它是通过从$S^{2} \times S^{2}$的标准细胞结构的单纯细分的广义bistellar移动。这两种结构都与二十面体有着惊人的密切关系。众所周知，$\CC P^{2}$有$S^{2} \times S^{2}$作为一个双重分支覆盖，我们构造了三角$\CC P^{2}$的$\CC P^{2}_{10}$通过提出一个简单的实现这个覆盖映射$S^{2} \times S^{2} \to \CC P^{2}$。该单纯映射的域是$S^{2} \times S^{2}$的标准单元结构的单纯细分，不同于（b）中提到的三角剖分。这给出了Kuehnel的$\CC P^{2}_{9}$三角形化$\CC P^{2}$的一个新的证明。$\CC P^{2}_{10}$和$(S^{2} \times S^{2})_{12}$分别在$\CC P^{2}$和$S^{2} \times S^{2}$上诱导出标准的分段线性结构.
英文摘要：
We present two constructions in this paper: (a) A 10-vertex triangulation $\CC P^{2}_{10}$ of the complex projective plane $\CC P^{2}$ as a subcomplex of the join of the standard sphere ($S^{2}_4$) and the standard real projective plane ($\RR P^{2}_{6}$, the decahedron), its automorphism group is $A_4$; (b) a 12-vertex triangulation $(S^{2} \times S^{2})_{12}$ of $S^{2} \times S^{2}$ with automorphism group $2S_5$, the Schur double cover of the symmetric group $S_5$. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that $\CC P^{2}$ has $S^{2} \times S^{2}$ as a two-fold branched cover; we construct the triangulation $\CC P^{2}_{10}$ of $\CC P^{2}$ by presenting a simplicial realization of this covering map $S^{2} \times S^{2} \to \CC P^{2}$. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$, different from the triangulation alluded to in (b). This gives a new proof that Kuehnel's $\CC P^{2}_{9}$ triangulates $\CC P^{2}$. It is also shown that $\CC P^{2}_{10}$ and $(S^{2} \times S^{2})_{12}$ induce the standard piecewise linear structure on $\CC P^{2}$ and $S^{2} \times S^{2}$ respectively.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
