Permuted difference cycles and triangulated sphere bundles Original Research Article

For any dimension d and any k = 1, …, d we construct a 2-neighborly triangulation of a d-manifold Mkd which is invariant under the action of the dihedral group Dn on n = 2d−k (k + 3) − 1 vertices. Mkd is the boundary of a (d + 1)-manifold Mkd + 1 with the same properties. Special cases in this family have been observed before: Mdd is the boundary of a (d + 1)-simplex, Md − 1d + 1 is an orientable or nonorientable 1-handle depending on the parity of d, M1d is a d-dimensional torus. Topologically, Mkd (or Mkd + 1) is the total space of a sphere bundle (or disc 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 (or d + 1) integers.