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). As an application we obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R^3 PL-equivalent to a dual manifold immersion of a cubical 4-polytope. 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. Our explicit example has 17 718 vertices and 16 533 facets. Thus we get a parity changing operation for 3-dimensional cubical complexes (hexa meshes); this solves problems of Eppstein, Thurston, and others.