A cubical 4-polytope with a dual Klein bottle.

A d-dimensional polytope is *cubical*
if all its facets are combinatorially
isomorphic to the (d-1)-dimensional standard
cube; compare [1, Sect. 4.6].

It has been
observed by Stanley and by MacPherson that every
cubical d-polytope *P* determines a
normal crossing codimension one PL immersion
of an abstract cubical (d-2)-manifold into
(the barycentric subdivision of) the boundary
of the polytope: Each vertex of the dual
manifold corresponds to an edge of *P*
and each facet separates two opposite faces of
a facet of *P*.

In the case
a of cubical 4-polytope each
connected component of the dual manifold is a
cubical surface (compact 2-manifold without
boundary).
The immersed manifold is orientable if and
only if the 2-skeleton of the cubical d-polytope is *``edge orientable''* in the sense of Hetyei, that is, there is no
orientation of the edges such that in each 2-face opposite
edges are parallel.
Hetyei conjectured that there are cubical 4-polytopes that are
not edge-orientable [2, Conj. 2].

In [3] we obtain the first instance of a
cubical 4-polytope (with 72 vertices and 62 facets) for which
the immersed dual surface is not orientable: One of its components
is a *Klein bottle*. This confirms Hetyei's conjecture.

This instance is constructed as follows (we refer to [3] for the details):

- Start with two copies of the
cubical octahedron (the only cubical
3-polytope with 8 facets) and glue them
together such that the outcome is a
polytopal 3-ball (with 2 facets) whose
2-skeleton contains a
*Möbius strip with parallel inner edges*. - Both facets of this polytopal 3-ball
are subdivided into cubes by the
*Schlegel cap*construction [3]. For each of the two copies of the cubical octahedron the Schlegel cap yields a regular cubical 3-ball isomorphic to the prism over a Schlegel diagram of the cubical octahedron. The Möbius strip is not affected by the subdivision. - A prism construction yields a cubical 4-polytope with 72 vertices and 62 facets. One component of its immersed dual manifold is a Klein bottle.

Model produced with: polymake 1.5.1 + additional tools

Keywords | cubical complexes; cubical polytope; regular subdivision; dual surface immersion; Klein bottle | |

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

Zentralblatt No. | 05264899 |

- B. Grünbaum: Convex Polytopes, Graduate Texts in Math. 221, Springer-Verlag, New York. Second edition by V. Kaibel, V. Klee and G. M. Ziegler (2003; original edition: Interscience, London 1967).
- Garbor Hetyei:
*On the Stanley ring of a cubical complex*, Discrete Comput. Geom.**14**, 3 (1995), 305-330. - Alexander Schwartz and Günter M. Ziegler:
*Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds*(2003), Preprint, http://arxiv.org/abs/math.CO/0310269.

- Master File: C4P_Klein_Master.poly
- Applet File: C4P_Klein_Applet.jvx
- Preview: C4P_Klein_Preview.gif
- Print File: C4P_Klein_Print.eps

Submitted: Fri May 21 18:25:54 CEST 2004.

Revised: Fri Nov 19 15:19:33 CET 2004.

Accepted: Mon Nov 29 13:51:04 MET 2004.

TU BerlinGünter M. Ziegler

Inst. Mathematics, MA 6-2

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