Year: 2025
Type: article
Abstract: According to a well-known result in geometric topology, we have $$\left( \mathbb {S}^2 \right) ^{n}\!\!/\operatorname {Sym}(n) = \mathbb{C}\mathbb{P}^{n}$$ S 2 n / Sym ( n ) = C P n , where $$\operatorname {Sym}(n)$$ Sym ( n ) acts on $$\left( \mathbb {S}^2 \right) ^{n}$$ S 2 n by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $$\mathbb{C}\mathbb{P}^{n}$$ C P n for each $$n \ge 2$$ n ≥ 2 . In more detail, we start with the standard two triangle crystallisation $$S^2_3$$ S 3 2 of the 2-sphere $$\mathbb {S}^2$$ S 2 , in its n -fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the $$\operatorname {Sym}(n)$$ Sym ( n ) quotient of this subdivision yields a simplicial cell decomposition of $$\mathbb{C}\mathbb{P}^n$$ C P n . The first derived subdivision of this cell complex is a simplicial triangulation of $$\mathbb{C}\mathbb{P}^n$$ C P n . To the best of our knowledge, this is the first explicit description of triangulations of $$\mathbb{C}\mathbb{P}^n$$ C P n for $$n \ge 4.$$
Authors: Basudeb Datta, Jonathan Spreer
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416804744