## Projectivities in simplicial complexes and colorings of simple polytopes

Venue: | Math. Z |

Citations: | 15 - 5 self |

### BibTeX

@ARTICLE{Joswig_projectivitiesin,

author = {Michael Joswig},

title = {Projectivities in simplicial complexes and colorings of simple polytopes},

journal = {Math. Z},

year = {},

pages = {10--1007}

}

### OpenURL

### Abstract

For each strongly connected finite-dimensional (pure) simplicial complex ∆ we construct a finite group Π(∆), the group of projectivities of ∆, which is a combinatorial but not a topological invariant of ∆. This group is studied for simplicial manifolds and, in particular, for polytopal simplicial spheres. The results are used to solve a coloring problem for simplicial (or, dually, simple) polytopes which arose in the area of toric algebraic varieties. 1

### Citations

817 | Introduction to graph theory - West - 2001 |

450 |
Elements of algebraic topology
- Munkres
- 1993
(Show Context)
Citation Context ... topology, everysnite-dimensional simplicial complex denes a locally compact and metrizable Hausdor space jjjj, which is compact if and only if issnite; see any topology textbook, e.g. Munkres [16], for the details. We frequently apply notions from topology to which, if no confusion can arise, are meant to refer to jjjj. The dual graph () of is an abstract graph whose nodes are the facets... |

359 |
Lectures on polytopes
- Ziegler
- 1995
(Show Context)
Citation Context ..., equivalently, for any given vertex v there is a 1-1 correspondence between the sets of edges through v and the faces containing v. For an introduction to the theory of convex polytopes, see Ziegler =-=[23-=-]. Here we restrict our attention to polytopes which are convex. There is another way to characterize simple polytopes, which suits our needs: A polytope P is simple if and only if its dual P is simp... |

109 |
Convex polytopes, Coxeter orbifolds and torus actions
- Davis, Januszkiewicz
- 1991
(Show Context)
Citation Context ...articular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric manifolds. 1 Introduction In =-=[6-=-] Davis and Januszkiewicz introduce an (n+d)-dimensional smooth manifoldsZ P built from a d-dimensional simple convex polytope P with n facets. These manifolds play a signicant role in the study of (q... |

68 |
Shellable decompositions of cells and spheres
- Bruggesser, Mani
- 1971
(Show Context)
Citation Context ...ructible if the intersection A \ B is a pure constructible complex. The notion of constructibility generalizes the concept of shellability, see Ziegler [23, x8]. From a theorem of Bruggesser and Mani =-=[-=-3] it is known that the boundary complexes of polytopes are shellable and thus constructible. The 1-skeleton of a polytopal complex forms an abstract graph (). For being the boundary of a convex poly... |

57 | The Four Color Problem - Ore - 1967 |

50 | Combinatorics and commutative - Stanley - 1996 |

46 |
Piecewise Linear Topology
- Hudson
- 1969
(Show Context)
Citation Context ...y, a PL-manifold M always admits a triangulation (compatible with the PL-structure), such that is a combinatorial manifold. For a general introduction to combinatorial and PL-manifolds see Hudson [1=-=-=-2], Glaser [10], or [13, 65 (IX.17)]. Throughout the following let be a combinatorial manifold. This implies that the dual graph () is strongly connected, so the isomorphism class of the group of pro... |

23 | Maldeghem, Generalized Polygons - Van - 1998 |

22 | Tangential structures on toric manifolds, and connected sums of polytopes
- Buchstaber, Ray
(Show Context)
Citation Context ...uivalent to any zonotope. The polytope M is constructed as the blending of two cubes. This operation on simple polytopes has been introduced by Barnette [1] as joining ; it is called connected sum in =-=[5]-=-. The f-vector of M equals (14; 21; 9). This is not the f-vector of any zonotope, because zonotopes, being centrally symmetric, have an even number of facets. One can show that among all even simple 3... |

19 |
Geometric combinatorial topology
- Glaser
- 1970
(Show Context)
Citation Context ...old M always admits a triangulation (compatible with the PL-structure), such that is a combinatorial manifold. For a general introduction to combinatorial and PL-manifolds see Hudson [12], Glaser [1=-=-=-0], or [13, 65 (IX.17)]. Throughout the following let be a combinatorial manifold. This implies that the dual graph () is strongly connected, so the isomorphism class of the group of projectivities d... |

4 | Torus actions, combinatorial topology and homological algebra
- Buchstaber, Panov
(Show Context)
Citation Context ...otient st 1 in the group T is trivial for all facets F i not containing the point p = q. We obtain a manifold Z P as the quotient space (P T )=. For a survey on the subject see Buchstaber and Panov [4=-=]-=-, where the construction of the manifold Z P is discussed in Section 3.1. The obvious action of the torus T on Z P is free over the interior of P . Points which are contained in the relative interior ... |

4 |
Nonpositive curvature of blow-ups, Sel
- Davis, Januszkiewicz, et al.
- 1998
(Show Context)
Citation Context ... M. Ziegler for giving helpful comments on a previous version of this paper. Moreover, I am grateful to the anonymous referee for bringing to my attention the paper of Davis, Januszkiewicz, and Scott [7]. 2 2 Simplicial Complexes An (abstract) simplicial complex on the vertex set V is a non-empty collection ofsnite subsets of V , which is closed with respect to forming subsets. If 2 with # = k... |

4 |
An amusing reformulation of the four color problem
- Edwards
- 1977
(Show Context)
Citation Context ...l. The result for the special case of simple zonotopes is implicit in [7, Lemma 4.2.6] of Davis, Januszkiewicz, and Scott. Moreover, since the original submission of this paper I learned that Edwards =-=[8]-=- had announced a solution to the coloring problem already in 1977. However, to the best of my knowledge, no proof was published. The paper is organized as follows. We start by associating asnite group... |

3 |
A simple 4-dimensional nonfacet
- Barnette
- 1969
(Show Context)
Citation Context ...le 3-polytope M which is not combinatorially equivalent to any zonotope. The polytope M is constructed as the blending of two cubes. This operation on simple polytopes has been introduced by Barnette =-=[1]-=- as joining ; it is called connected sum in [5]. The f-vector of M equals (14; 21; 9). This is not the f-vector of any zonotope, because zonotopes, being centrally symmetric, have an even number of fa... |

3 |
and Hironori Onishi, Even triangulations of S 3 and the coloring of graphs, Trans
- Goodman
- 1978
(Show Context)
Citation Context ... proof. The proofs employ techniques, for which it seems to be unclear how they can be generalized to higher dimensions. The result for 4-dimensional polytopes follows from work of Goodman and Onishi =-=[11]-=-. Davis, Januszkiewicz, and Scott proved in [7, Lemma 4.2.6] that the boundary complex of the dual of a simple zonotope is balanced. Figure 4: Even simple 3-polytope M which is not combinatorially equ... |

2 | Billera and Anders Björner, Face numbers of polytopes and complexes - Louis - 1997 |

2 | A Textbook of Topology. Transl. by Michael - Seifert, Threlfall - 1980 |

1 |
and Hironori Onishi, Even triangulations of S and the coloring of graphs
- Goodman
- 1978
(Show Context)
Citation Context ... proof. The proofs employ techniques, for which it seems to be unclear how they can be generalized to higher dimensions. The result for 4-dimensional polytopes follows from work of Goodman and Onishi =-=[11]-=-. Davis, Januszkiewicz, and Scott proved in [7, Lemma 4.2.6] that the boundary complex of the dual of a simple zonotope is balanced. Figure 4: Even simple 3-polytope M which is not combinatorially equ... |

1 |
private communication
- Izmestiev
- 2001
(Show Context)
Citation Context ... from 0 [ 0 0 onto itself can be written as the product = h 0 [ 0 0 ; 1 [ 0 0 ; : : : ; ( m = 0 ) [ 0 0 )i h 0 [ 0 0 ; 0 [ 0 1 ; : : : ; 0 [ ( 0 n = 0 0 )i: Izmestiev [14] has proved a partial converse of the previous proposition. The d-dimensional simplicial complex on the vertex set V is called balancedsif there is a map c : V ! f0; : : : ; dg such that whenever fv... |