On the genus of the complex projective plane (Q1120857)

The paper is devoted to the construction of a pseudo-triangulation of the complex projective plane \({\mathbb{C}}P^ 2\), with 5 vertices, 10 edges, 20 triangles, 20 tetrahedra and 8 5-simplexes, which is proved to be minimal with respect to both the number of vertices and 5-simplexes. Moreover, the associated crystallization (of order 8) admits a regular imbedding into the closed orientable surface of genus two. Since \({\mathbb{S}}^ 4\) is the only 4-manifold of regular genus zero and \({\mathbb{S}}^ 1\times {\mathbb{S}}^ 3\) is the only orientable 4-manifold of regular genus one (as proved by \textit{M. Ferri} and the author [Proc. Am. Math. Soc. 85, 638-642 (1982; Zbl 0522.57021)] and by \textit{A. Cavicchioli} [Proc. Am. Math. Soc. 105, 1008-1014 (1989)], respectively), then the genus of \({\mathbb{C}}P^ 2\) is proved to be two. As a consequence, the genus of the twisted \({\mathbb{S}}^ 2\)-bundle over \({\mathbb{S}}^ 1\) is proved to be less or equal than four.