To any finite poset P we associate two graphs which we denote by Ω(P ) and 0(P ). Several standard constructions can be seen as Ω(P ) or 0(P ) for suitable posets P , including the comparability graph of a poset, the clique graph of a graph and the 1–skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we...

In this article a generalized version of small cancellation theory is developed which is applicable to specific types of high-dimensional simplicial complexes. The usual results on small cancellation groups are then shown to hold in this new setting with only slight modifications. For example, arbitrary dimensional versions of the Poincaré construction and the Cayley complex are described.

The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Ax<~ b), with A a fixed generic (n + 1 ) × n matrix. The topological space associated with K(A) is shown to be homeomorphic to R n, and the space obtained by identifying lattice translates of these simplices is homeorphic to the n-toms.

The notion of k-systolic (k ≥ 6 being a natural number) complexes was introduced by T. Januszkiewicz and J. Świa̧tkowski [JS1] and, independently, by F. Haglund [H] as combinatorial analogue of nonpositively curved spaces. Those complexes are simply connected simplicial complexes satisfying some local combinatorial conditions. Roughly speaking there is a lower bound for the length of “essential"...

We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen’s generalized lower bounds, and of Barnette’s lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.

Abstract simplicial complexes are used in many application contexts to represent multi-dimensional, possibly non-manifold and nonuniformly dimensional, geometric objects. In this paper we introduce a new general yet compact data structure for representing simplicial complexes, which is based on a decomposition approach that we have presented in our previous work [3]. We compare our data structu...

We study the maintenance of a simplicial grid or complex under changing density requirements. The proposed method works in any xed dimension and generates grids by projecting cross-sections of a monotone simplicial complex that lives in one dimension higher than the grid. The density of the grid is adapted by locally moving the cross-section up or down along the extra dimension.

We show that each polynomial a(z)=1+a1z+· · ·+adzd inN[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen–Macaulay ring and is the g-polynomial of a simplicial polytope.We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zero...

We establish a new homological lower bound for the Thurston norm on 1-cohomology of 3-manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1-cohomology of 2-dimensional CW-complexes. AMS Classification 57M27, 57M20, 57M05

Orbispaces are spaces with extra structure. The main examples come from compact Lie group actionsX G and are denoted [X/G], their underlying space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. To be more precise, an orbispace is a topological stack which is locally equivalent to [X/G] for G a Lie group and X a G-CW-complex [2] [4] [5]. Howev...