From the icosahedron to natural triangulations of \(\mathbb {C}\mathbb{P}^{2}\) and \(S ^{2}\times S ^{2}\). (English) Zbl 1231.51016
The authors present two constructions: a 10-vertex triangulation of the complex projective plane (as a subcomplex of the join of the sphere and the real projective plane), and a 12-vertex triangulation of \(S^2 \times S^2\). The automorphism group is \(A_4\) in the first construction and the Schur double cover of \(S_5\) in the second construction. Both constructions have an intimate relationship with the icosahedron, in some sense analogous to the (well-known) fact that the minimal triangulation of the real projective plane arises naturally from the icosahedron.
Reviewer: Matthias Beck (San Francisco)
MSC:
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 52B10 | Three-dimensional polytopes |
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