A Triangulation of ℂP 3 as Symmetric Cube of S 2

The symmetric group S3 acts on S 2 × S 2 × S 2 by coordinate permutation, and the quotient space (S 2×S 2×S )/S3 is homeomorphic to the complex projective space CP 3. In this paper, we construct an 124-vertex simplicial subdivision (S 2 ×S 2 ×S )124 of the 64vertex standard cellulation S 2 4 × S 2 4 × S 2 4 of S 2 × S 2 × S 2, such that the S3-action on this cellulation naturally extends to an action on (S 2 × S 2 × S )124. Further, the S3-action on (S 2×S 2×S )124 is “good”, so that the quotient simplicial complex (S 2×S 2×S )124/S3 is a 30-vertex triangulation CP 3 30 of CP 3. In other words, we construct a simplicial realization (S 2 × S 2 × S )124 → CP 3 30 of the branched covering S 2 × S 2 × S 2 → CP 3. Finally, we apply the BISTELLAR program of Lutz on CP 3 30, resulting in an 18-vertex 2-neighbourly triangulation CP 3 18 of CP 3. The automorphism group of CP 3 18 is trivial. It may be recalled that, by a result of Arnoux and Marin, any triangulation of CP 3 requires at least 17 vertices. So, CP 3 18 is close to vertex-minimal, if not actually vertex-minimal. Moreover, no explicit triangulation of CP 3 was known so far. MSC 2010: 57Q15, 57R05, 57M60.