Abstract According to a well-known result in geometric topology, we have (S²) ^n\!\!/ Sym (n) = CP^n 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 CP^n 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.
Datta et al. (Fri,) studied this question.