From the icosahedron to natural triangulations of $\CC P^2$ and $S^2\times S^2$

Abstract

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 ($S_4^2$) 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.

18 pages, revised version, to appear in `Discrete & Computational Geometry'

References in corpus (2)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Exotic smooth structures on $S^2\times S^2$

Cited by in corpus (1)

• Combinatorics of embeddings