Hamiltonian Submanifolds of Regular Polytopes

This work is set in the field of combinatorial topology, a mathematical field of research in the intersection of the fields of topology, geometry, polytope theory and combinatorics. This work investigates polyhedral manifolds as subcomplexes of the boundary complex of a regular polytope. Such a subcomplex is called k-Hamiltonian, if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), the focus here is on the case of the cross polytope and the sporadic regular 4-polytopes. By the results presented, the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore, 2Hamiltonian 4-manifolds in the d-dimensional cross polytope are investigated. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, a new example with 16 vertices which is highly symmetric with an automorphism group of order 128 is presented. Topologically, it is homeomorphic to a connected sum of 7 copies of S2 S2. By this example all regular cases of n vertices with n @ 20 or, equivalently, all cases of regular d-polytopes with d B 9 are now decided. The notion of tightness of a PL-embedding of a triangulated manifold is closely related to its property of being a Hamiltonian subcomplex of some convex polytope. Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplex-wise linear embedding of the triangulation into Euclidean space is “as convex as possible”. It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here, we present other sufficient and purely combinatorial conditions which can be applied