O ct 2 00 3 Construction techniques for cubical complexes , odd cubical 4 - polytopes , and prescribed dual manifolds

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). Thus we obtain the first instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms the existence conjecture of Hetyei [17, Conj. 2, p. 325]. More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R is PL-equivalent to a dual manifold immersion of a cubical 4polytope. As an instance we obtain a cubical 4-polytope with a cubation of Boy’s surface as a dual manifold immersion, and with an odd number of facets. This solves problems of Eppstein, Thurston and others. Our explicit example has 19 520 vertices and 18 333 facets.