Figure 1: A 4-dimensional realization. |

Figure 2: The linked link of vertex A. |

Figure 3: A Z-acyclic triangulation. |

We present a 4-dimensional polyhedral realization of a 2-dimensional Z-acyclic but non-contractible simplicial complex with 23 vertices.

Our example answers a query by Lutz Hille (Hamburg), who in
November 2006 had asked us for examples
of Z-acyclic but non-contractible complexes
realized in low dimensions. His question was motivated by
toric geometry. Indeed, any 2-dimensional complex with rational
coordinates in **R**^{4}
corresponds to a subfan of a complete fan in **R**^{5},
which in turn determines a 5-dimensional toric variety. Cohomology
computations for line bundles on the toric variety, as considered in [1, 7, 8],
can be reduced to cohomology computations for closed subcomplexes,
as provided here. In this context, the question for the difference
between homological triviality (acyclicity) and homotopical
triviality (contractibility) arises naturally.

Our model demonstrates explicitly that for 2-dimensional complexes
piecewise linearly (PL) embedded in **R**^{4}
homotopy conditions are stronger than homological ones.
The respective dimensions hereby are smallest possible:
Z-acyclic but non-contractible 2-dimensional complexes
do not have PL embeddings in **R**^{3},
whereas all 1-dimensional (finite) Z-acyclic complexes are contractible.

A famous example of a Z-acyclic but non-contractible complex
is due to Floyd and Richardson [4]: Let SO(3) be the group of
3-dimensional rotations and I be the subgroup of rotational
symmetries of the icosahedron. The group I is isomorphic to
the alternating group A_{5} and acts on the coset space SO(3)/I,
which is homeomorphic to the Poincaré homology 3-sphere; cf. [10].
By removing the open star of the point I from (an appropriate
cell decomposition of) the quotient SO(3)/I, we obtain
a 2-dimensional (cellular) complex K that is Z-acyclic
but non-contractible and on which I acts freely.
Alternatively, the complex K arises as the 2-skeleton
of the Poincaré homology 3-sphere in its description
by Weber and Seifert [17] as the spherical dodecahedral space.
The complex K has the binary icosahedral group I^{*} as its fundamental group.

In fact, every homology 3-sphere Σ^{3} different from S^{3}
can be used to yield a Z-acyclic but non-contractible 2-dimensional complex.
One way to describe any such Σ^{3} is as a 3-dimensional (polyhedral) ball
with identifications on the boundary (cf. Seifert [16]). As in the Floyd-Richardson example,
the identified boundary is a Z-acyclic but non-contractible 2-dimensional complex.

Newman [15] gave yet another construction of a Z-acyclic complex (non-homeomorphic
to the complex K above, but also with fundamental group I^{*})
by identifying the boundaries of two circular discs. Wedges as well as connected sums
of Z-acyclic complexes are Z-acyclic again and thus yield further examples.

All finite 1-dimensional Z-acyclic complexes are trees and hence are contractible.
As planar graphs, they are geometrically realizable in **R**^{2}.

The embedding respectively the PL embedding of 2-dimensional Z-acylic complexes
is a more delicate matter. As pointed out by Zeeman [19], there are contractible
2-dimensional simplicial complexes that cannot be embedded (topologically)
in any 3-manifold, for example, cones over non-planar graphs.
However, any (finite) 2-complex is embeddable in some 4-manifold
and every contractible 2-complex can be embedded in **R**^{4}; see Curtis [2].
(As remarked in [12], it is not known whether all contractible
2-dimensional complexes have a PL embedding in **R**^{4}.)
Such embeddings can be complicated (Curtis, cf. [5]): there is an embedding
of a 2-dimensional contractible complex L in S^{4} such that the complement
S^{4}\L is not simply connected. Note that in this situation
Alexander duality implies that S^{4}\L is Z-acyclic.
For further embeddings with non-simply connected complement see Glaser [5, 6]
and Neuzil [14].

It was later proved by Kranjc [11; Cor. 2 and Rem. 2] that arbitrary finite Z-acylic
2-dimensional complexes are topologically embeddable in **R**^{4}.
A class of polyhedral homology 3-spheres in **R**^{4}
with non-trivial fundamental group was given by Curtis and Wilder [3].
These yield examples of PL embeddings of non-contractible Z-acylic
2-dimensional complexes in **R**^{4}.
In the following, we present a model with few vertices.

**Theorem:**
There is a 2-dimensional Z-acyclic but non-contractible simplicial complex on 23 vertices
that has a polyhedral realization (with small integer coordinates) in **R**^{4}.

*Proof:*
We appropriately subdivide the 2-skeleton of the spherical dodecahedral space
(see above and cf. [13]) to obtain a Z-acyclic but non-contractible 2-dimensional
simplicial complex with 23 vertices, A, B,...,W;
see Figure 3 for the respective triangulation.
We then remove the star of the vertex A from the triangulation
to obtain the shaded complex of Figure 3.
A 3-dimensional geometric realization of this shaded complex is visualized in Figure 1.
Finally, the star of vertex A is added by placing this vertex
in **R**^{4} at position (0,0,0,1). To be more precise:
The vertices B, C, D, and E of the shaded complex are connected
to each other. They form the complete graph K_{4}
which is the 1-skeleton of the tetrahedron. Each of the six
original pentagons of the 2-skeleton of the spherical dodecahedral space
is subdivided (in a symmetric way) by inserting three additional vertices each.
For example, the pentagon A-B-C-D-E is subdivided by adding the vertices F, G, and H.
In each of the six pentagons there is a strip of five shaded triangles
which is glued to the tetrahedral graph formed by the vertices
B, C, D, and E. Each of the stripes is colored in a different color.
It can be inspected in the applet how the stripes wind arround
the tetrahedral graph. The coordinates of the realization are chosen
such that the rotation group of the tetrahedron acts on the model.
All the coordinate entries that occur in the model are integers between -4 and +4.

The 3-dimensional realization of the shaded complex cannot be extended to a 3-dimensional realization of the full complex since the link of vertex A is linked in the realization (see Figure 2).

In fact, 2-dimensional Z-acyclic but non-contractible simplicial complexes
never have PL embeddings in **R**^{3}: Suppose, there is
such a PL embedding. Any regular neighborhood (cf. [18]) of the embedded
complex yields a Z-acyclic 3-manifold with **S**^{2} as boundary (cf. [9, p. 114]).
However, by the Schoenflies theorem, **S**^{2} bounds a 3-ball
with trivial fundamental group, contradiction.

Acknowledgement: The authors were supported by the DFG Research Group "Polyhedral Surfaces", Berlin.

Keywords | Z-acyclic and contractible complexes; polyhedral realizations | |

MSC-2000 Classification | 52B99 (57Q15, 51M20) | |

Zentralblatt No. | 05613404 |

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- Master File: Z_acyclic_2d_in_4d.jvx
- Applet File: Z_acyclic_2d_in_4d.jvx
- Applet File: Z_acyclic_2d_in_4d_edges.jvx
- Preview: Z_acyclic_2d_in_4d.gif
- Preview: Z_acyclic_2d_in_4d_edges.gif
- Image: Z_acyclic_2d.gif

Submitted: Wed May 30 09:47:38 CEST 2007.

Revised: Fri Nov 21 15:05:31 CET 2008.

Accepted: Mo Nov 24 15:28:50 CET 2008.

Technische Universität BerlinGünter M. Ziegler

Institut für Mathematik, MA 3-2

10623 Berlin

Germany

lutz@math.tu-berlin.de

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

Technische Universität Berlin

Institut für Mathematik, MA 6-2

10623 Berlin

Germany

ziegler@math.tu-berlin.de

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