Simplicial cell decompositions of CP^.3mmn

According to a well-known result in geometric topology, we have ( S² ) ⁿ\!\!/Sym(n) = CPⁿ S 2 n / Sym ( n ) = C P n , where Sym(n) Sym ( n ) acts on ( S² ) ⁿ S 2 n by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of CPⁿ C P n for each n ≥ 2 n ≥ 2 . In more detail, we start with the standard two triangle crystallisation S²₃ S 3 2 of the 2-sphere S² S 2 , in its n -fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the Sym(n) Sym ( n ) quotient of this subdivision yields a simplicial cell decomposition of CPⁿ C P n . The first derived subdivision of this cell complex is a simplicial triangulation of CPⁿ C P n . To the best of our knowledge, this is the first explicit description of triangulations of CPⁿ C P n for n ≥ 4.