Hamiltonian Submanifolds of Regular Polytopes

Abstract

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex 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 superneighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S 2 × S 2 . By this example all regular cases of n vertices with n < 20 or, equivalently, all cases of regular d-polytopes with d ≤ 9 are now decided.

Centrally symmetric manifolds with few vertices

“…In his Doctoral thesis [16], Sparla constructed a cs 12-vertex triangulation of S 2 × S 2 , see also [7], and conjectured that there exists a cs 4k-vertex triangulation of S k−1 × S k−1 for every k. Lutz [10], with an aid of computer programs MANIFOLD VT and BISTELLAR, confirmed this conjecture for k = 4 and k = 5 as well as found many cs 2d-vertex triangulations of S i × S d−i−2 for d ≤ 10. Very recently, Effenberger [2] proposed a certain construction of cs simplicial complexes with 4k vertices that conjecturally triangulate S k−1 × S k−1 ; with the help of the software package simcomp he then verified that this indeed holds for all values of k ≤ 12, thus establishing Sparla's conjecture up to k = 12.…”

Stacked polytopes and tight triangulations of manifolds

“…Proof. The triangulation (S 2 ×S 2 ) #7 16 is a combinatorial manifold and a tight subcomplex of β 8 as shown in [23]. Thus, each vertex link is a PL 3-sphere.…”

Highly neighborly centrally symmetric spheres

“…In this section, we sketch Jockusch's construction [6] of cs combinatorial 3-spheres with 2n vertices, ∆ 3,2 n , that are cs-2-neighborly. To prepare ground for our general construction we use notation different from the one used in [6].…”

“…= ∂C * 4 as the initial complex. Assume that inductively we constructed a cs 3-sphere ∆ 3,2 n that is cs-2-neighborly, has vertex set V n = {±v 1 , . .…”

“…A few observations are in order. First, by property (P n ), B 3,1 n is a subcomplex of ∆ 3,2 n , and hence by symmetry so is −B 3,1 n . Second, by definition of B 3,1 n , the subcomplexes B 3,1 n and −B 3,1 n share no common facets and…”