Higherdimensional analogues of Császár's torus
For every even dimension n we give an example of a \((2n+3)\)-vertex 2- neighborly combinatorial manifold which is a triangulation of the sphere product \(S^{n-1}\times S^ 1.\) It can be realized in the n-skeleton of the cyclic polytope \(C(2n+3,n+2)\) and consequently it is polyhedrally realizable as a hypersurface in euclidean \((n+1)-\)space. Because the triangulation contains all possible edges this polyhedron has no diagonals. In the case \(n=2\) this is known as Császár's torus. The same construction works for odd n leading to a \((2n+3)-\)vertex 2- neighborly triangulation of the (nonorientable) ''generalized Klein bottle'' which cannot be embedded into \((n+1)-\)space. However, it can be realized in the n-skeleton of the cyclic polytope \(C(2n+3,n+3)\) and consequently in euclidean \((n+2)-\)space. In the case \(n=3\) this triangulation has been found originally by \textit{A. Altshuler} and \textit{L. Steinberg} who called it \(N^ 9_{51}\) [Discrete Math. 8, 113-137 (1974; Zbl 0292.57011)].
scientific article; zbMATH DE number 3879631
scientific article