Equilibrium Triangulations of the Complex Projective Plane
by Thomas F. Banchoff and W. Kuhnel

By using the familiar 7-vertex triangulation of the torus, we contruct the 10-vertex triangulation of CP^2 that fits the equilibrium decomposition of CP^2 in the simplest way possible. We then exhibit the full automorphism group of order 42 by examining a finite group of Fubini-Study isometries. This leads to a proof of the Kuiper-Massey's theorem (which states that the standard 4-sphere is PL homeomorhic to CP^2 modulo conjugation. Finally, the tight simplicial embeddings of CP10^2 and CP7^2 are derived.