Permuted difference cycles and triangulated sphere bundles
As a generalization of the classical 7-vertex torus, a 2-parameter family of 2-neighborly triangulated \(d\)-manifolds \(M^d_k\) is constructed, each invariant under the action of the dihedral group \(D_n\) on \(n = 2^{d - k} (k + 3) - 1\) vertices. \(M^d_k\) occurs as the boundary of a \((d + 1)\)-manifold with the same properties. In particular, \(M^d_d\) is the boundary of a \((d + 1)\)-simplex, \(M^d_{d - 1}\) is the boundary of an orientable or nonorientable 1-handle depending on the parity of \(d\), \(M^d_1\) is a \(d\)-dimensional torus. Topologically \(M^d_k\) is the total space of a sphere bundle over a \((d - k)\)-dimensional torus. The construction of the triangulation itself is purely combinatorial. It is based on permutations of certain difference cycles encoding all the information about the triangulation in \(d\) integers.
scientific article; zbMATH DE number 812489
scientific article