The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general suffici...

We compute the Euler-Poincaré characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use this result to determine the structure of the quotient simplicial complex in certain cases.

Some recent attempts of settling this question go by looking at the problem in the more general context of pure simplicial complexes: What is the maximum diameter of the dual graph of a simplicial (d − 1)sphere or (d− 1)-ball with n vertices? Here a simplicial (d − 1)-ball or sphere is a simplicial complex homeomorphic to the (d − 1)-ball or sphere. These complexes are necessarily pure (all the...

We exhibit an explicit homotopy equivalence between the geometric realizations of the order complex of a finite lattice and the simplicial complex of coreless sets of atoms whose join is not 1̂. This result, which extends a theorem of Segev, leads to a description of the homology of a finite lattice, extending a result of Björner for geometric lattices.

In this paper, we discuss the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves. One construction proceeds via Brown representability and provides a classification in the pointed model category. The second construction is given by the classifying space of the monoid of homotopy self-equivalences of a simplicial sheaf and provides the unpointed class...

For a simplicial complex ∆, the graded Betti number βi,j(k[∆]) of the StanleyReisner ring k[∆] over a field k has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if ∆ is the boundary complex of a d-dimensional stacked polytope with n vertices for d ≥ 3, then βk−1,k(k[∆]) = (k − 1) ` n−d k ́ . We prove this combinatorially.

In this paper we determine the threshold for d-collapsibility in the probabilistic model Xd(n, p) of d-dimensional simplicial complexes. A lower bound for this threshold p = ηd n was established in [?], and here we show that this is indeed the correct threshold. Namely, for every c > ηd, a complex drawn from Xd(n, c n ) is asymptotically almost surely not d-collapsible.

We give new examples of shellable but not extendably shellable two dimensional simplicial complexes. They include minimal examples, which are smaller than those previously known. We also give examples of shellable but not vertex decomposable two dimensional simplicial complexes. Among them are extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability ...

We discuss recursive formulas for Branson’sQ-curvatures. The formulas present Q-curvatures of any order in terms of lower order Qcurvatures and lower order GJMS-operators. These presentations are universal in the sense that the recursive structure does not depend on the dimension of the underlying space. We give proofs for Q4 and Q6 for general metrics, and for Q8 for conformally flat metrics. ...

The main result of the paper contains an exact formula for the rate of regularity of a set-valued mapping with a convex graph. As a consequence we find an exact expression for the rate of regularity of a set-valued mapping associated with so called constraint system. It turns out that the rate is equal to the upper bound of Robinson-type estimates over all norms in the graph space of the homoge...