, Volume 46, Issue 3, pp 542-560
Date: 24 Aug 2010

From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2

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Abstract

We present two constructions in this paper: (a) a 10-vertex triangulation \(\mathbb{C}P^{2}_{10}\) of the complex projective plane ℂP 2 as a subcomplex of the join of the standard sphere ( \(S^{2}_{4}\) ) and the standard real projective plane ( \(\mathbb{R}P^{2}_{6}\) , the decahedron), its automorphism group is A 4; (b) a 12-vertex triangulation (S 2×S 2)12 of S 2×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×S 2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ℂP 2 has S 2×S 2 as a two-fold branched cover; we construct the triangulation \(\mathbb{C}P^{2}_{10}\) of ℂP 2 by presenting a simplicial realization of this covering map S 2×S 2→ℂP 2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2×S 2, different from the triangulation alluded to in (b). This gives a new proof that Kühnel’s \(\mathbb{C}P^{2}_{9}\) triangulates ℂP 2. It is also shown that \(\mathbb{C}P^{2}_{10}\) and (S 2×S 2)12 induce the standard piecewise linear structure on ℂP 2 and S 2×S 2 respectively.

The research of B. Datta was supported by UGC grant UGC-SAP/DSA-IV.