Permuted difference cycles and triangulated sphere bundles (Q1901130)

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.