A cubical 4-polytope with an odd number of facets and a dual Boy's surface.

It has been observed by Stanley and by MacPherson that every cubical d-polytope determines a PL immersion of an abstract cubical (d-2)-manifold into (the barycentric subdivision of) the boundary of the polytope [1]. In the case of cubical 4-polytopes each connected component of the dual manifold is a surface (a compact 2-manifold without boundary).

We prove in [3] that *every* normal crossing
codimension one immersion of a compact
2-manifold into R^{3} is PL-equivalent
to a dual manifold immersion of a cubical
4-polytope. Thus, in particular non-orientable
dual 2-manifolds do arise: In [4] we described a rather small and
simple instance of a 4-polytope with
a non-orientable dual 2-manifold (one component is
a Klein bottle). If the dual two-manifold has odd
Euler characteristic (with an odd number of non-orientable components of odd genus),
the resulting cubical 4-polytope has an odd number of facets.

Here we present an example of a cubical 4-polytope with an odd number of facets, which has Boy surface (an immersed projective plane with a single triple point) as a dual manifold immersion. This solves problems of Eppstein, Thurston and others [2]. Our explicit example has 17,718 vertices and 16,533 facets. From this example, it follows in particular that every combinatorial cube has a subdivision into an even number of cubes (without subdividing the boundary). Moreover, it yields that for any geometric hexa mesh the flip graph (see Bern et al. [2]) has at least two connected components.

Our figures represents the key ideas for the construction: For a
given, small (74 vertices) lattice version of Boy's surface, one produces
a regular subdivision for a "pile of cubes" that has a subdivision of
the Boy surface as a dual manifold. The result (file `C4P_Boy_C3B_with_dual_Boy_surface.hexas`

)
is then lifted to a
4-polytope by a prism-type construction. (See [3] for details.)

The existence of cubical d-polytopes with an odd number of facets is governed by subtle topological data/obstructions. In particular, such polytopes exist for d=3 (easy) and d=4 (presented here), but not for d=6,8,9 or 10. The cases d=5 and d=7 remain open, for now.

Our submission includes the regular subdivision for a "pile of cubes" that has a subdivision of
the Boy surface as a dual manifold (file `C4P_Boy_C3B_with_dual_Boy_surface.hexas`

).
Furthermore, we provide our set of "templates" used to construct this regular cubical *3*-ball:
For each possible type of a vertex star of a normal-crossing grid immersion of a surface (compare the image files "`template_*_vertexstar.jpg`"), the set of templates
contains a regular cubical subdivision of the standard cube with a dual manifold PL-isomorphic to the vertex star
of the grid immersion. This set of templates can be used to produce cubical 4-polytopes with prescribed dual manifold immersion [3, Thm. 7.3]. (Consider the `C4P_Boy_readme.txt`

for a description of the file format.)

The cubicality of the 4-polytope `C4P_Boy_Master.poly`

can be verified using polymake 2.1 and polymake rules file `c4p.rules`

.

Model produced with: polymake 2.1

Keywords | cubical complexes; cubical polytopes; regular subdivision; normal crossing codimension one PL immersion; Boy's surface; grid immersion | |

MSC-2000 Classification | 52B12 (52B11, 52B05) | |

Zentralblatt No. | 05264900 |

- Eric K. Babson and Clara S. Chan:
*Counting faces for cubical spheres modulo two*, Discrete Math.**212**, 3 (2000), 169-183. - Marshall Wayne Bern, David Eppstein, and Jeffrey Gordon Erickson:
*Flipping cubical meshes*, Engineering with Computers**18**, 3 (2002), 173-187. - Alexander Schwartz and Günter M. Ziegler:
*Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds*, Experimental Mathematics**13**, 4 (2004), 385-413. - Alexander Schwartz and Günter M. Ziegler:
*A cubical 4-polytope with a dual Klein bottle*(2004), Electronic Geometry Model No. 2004.05.001, http://www.eg-models.de/2004.05.001.

- Master File: C4P_Boy_Master.poly
- Applet File: C4P_Boy_Fig1_Applet.jvx
- Applet File: C4P_Boy_Fig2_Applet.jvx
- Applet File: C4P_Boy_Fig3_Applet.jvx
- Preview: C4P_Boy_Fig1_Preview.jpg
- Preview: C4P_Boy_Fig2_Preview.jpg
- Preview: C4P_Boy_Fig3_Preview.jpg
- Readme File: C4P_Boy_readme.txt
- Other: C4P_Boy_C3B_with_dual_Boy_surface.hexas
- Other: template_empty_cubification.hexas
- Other: template_single3_cubification.hexas
- Other: template_single3_vertexstar.jpg
- Other: template_single4a_cubification.hexas
- Other: template_single4a_vertexstar.jpg
- Other: template_single4b_cubification.hexas
- Other: template_single4b_vertexstar.jpg
- Other: template_single5_cubification.hexas
- Other: template_single5_vertexstar.jpg
- Other: template_double8a_cubification.hexas
- Other: template_double8a_vertexstar.jpg
- Other: template_double8b_cubification.hexas
- Other: template_double8b_vertexstar.jpg
- Other: template_triple_cubification.hexas
- Other: template_triple_vertexstar.jpg
- Other: c4p.rules

Submitted: Tue Jul 13 11:37:01 CEST 2004.

Revised: Mon Jul 4 12:20:08 CEST 2005.

Accepted: Tue Jul 26 12:12:12 MET DST 2005.

TU BerlinGünter M. Ziegler

Inst. Mathematics, MA 6-2

Strasse des 17. Juni 136

10623 Berlin

Germany

schwartz@math.tu-berlin.de

http://www.math.tu-berlin.de/~schwartz

TU Berlin

Inst. Mathematics, MA 6-2

Strasse des 17. Juni 136

10623 Berlin

Germany

ziegler@math.tu-berlin.de

http://www.math.tu-berlin.de/~ziegler