Abstract
Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of ℂP2 which fits the equilibrium decomposition of ℂP2 in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that ℂP2 modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation ℝP27of the real projective plane. We also determine all tight simplicial embeddings of ℂP210and ℝP27.
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