In a previous paper [ibid. 46, 542–560 (2011; Zbl 1231.51016)] the authors constructed a 10-vertex triangulation of \(\mathbb{C} P^2\) as a branched quotient of the 16-vertex product \(S^2\times S^2\), subdivided in a certain way. This branched simplicial covering preserves a large part of the symmetry.
In the paper under review, they start with the 64-vertex product \(S^2\times S^2\times S^2\), subdivided in a certain way, and construct a 30-vertex simplicial triangulation of \(\mathbb{C}^3\) as a branched quotient. Here the subdivision requires 60 extra vertices since there is no simultaneous and coherent triangulation of all the various cubes that come in here. The mapping from \((S^2)^3\) to \(\mathbb{C} P^3\) is a simplicial branched covering, and again a large symmetry group is preserved. By bi-stellar moves, the number of vertices for \(\mathbb{C} P^3\) could be reduced to 18 but not (or not yet) to 17. There is no 16-vertex triangulation according to P. Arnoux and A. Marin [Mem. Fac. Sci., Kyushu Univ., Ser. A 45, No. 2, 1670–244 (1991; Zbl 0753.52002)].