A triangulation of P3 as symmetric cube of S2
Abstract
The symmetric group S3 acts on S2 × S2 × S2 by coordinate permutation, and the quotient space (S2 × S2 × S2)/S3 is homeomorphic to the complex projective space P3. In this paper, we construct an 124-vertex simplicial subdivision (S2 × S2 × S2)124 of the 64-vertex standard cellulation S24 × S24 × S24 of S2 × S2 × S2, such that the S3-action on this cellulation naturally extends to an action on (S2 × S2 × S2)124. Further, the S3-action on (S2 × S2 × S2)124 is "good", so that the quotient simplicial complex (S2 × S2 × S2)124/S3 is a 30-vertex triangulation P330 of P3. In other words, we construct a simplicial realization (S2 × S2 × S2)124 P330 of the branched covering S2 × S2 × S2 P3. Finally, we apply the BISTELLAR program of Lutz on P330, resulting in an 18-vertex 2-neighbourly triangulation P318 of P3. The automorphism group of P318 is trivial. It may be recalled that, by a result of Arnoux and Marin, any triangulation of P3 requires at least 17 vertices. So, P318 is close to vertex-minimal, if not actually vertex-minimal. Moreover, no explicit triangulation of P3 was known so far.