In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual.
Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that "cut across an ideal." Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular "generalized Bier spheres." In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties.
In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields "many shellable spheres", most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.
We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition is given so that this relationship is actually a subdivision. We use the results to generalize the proof method of Čukić and Delucchi, so far restricted to semilattices, for a result of Björner, Paffenholz, Sjöstrand and Ziegler on the Bier construction on posets.
The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier. Björner, Paffenholz, Sjöstrand and Ziegler generalize this construction to obtain a Bier poset Bier(P,I) from any bounded poset P and any proper ideal I of P. They show shellability of Bier(P,I) for the case where P is the boolean lattice, and obtain thereby 'many shellable spheres' in the sense of Kalai. We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov. We obtain 'more shellable spheres' by proving the general statement that combinatorial …
A Bier sphere $Bier(K) = K\ast_ΔK^\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K^\circ$, is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to $K$ a natural fan $Fan(K)$, related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify th…
Chromatic numbers, Buchstaber numbers and chordality of Bier spheres
Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics（数学物理力学研究所）
Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojević, Frick and Ziegler. The associated "Bier complexes" include both the Bier spheres and "optimal multiple chessboard complexes" as interesting, special cases. Our main result is Theorem 4.3 saying that (1) the $r$-fold deleted join of Alexander $r$-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the $r$-fold deleted join of a collectively unavoidable $r$-tuple is $(n-r-1)$-connected. We also give a comple…
We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicia…
We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $Π_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of spheres of dimension $(n-2)$. We extend on this result to show that $Π_{n,s}$ is edge-lexicographic shellable. We then use this edge-labeling to find a recursive expression for the number of spheres, and show that when $s=1$ the number of spheres is equal to the number of complete non-ambiguous trees, first defined in 2014 by Aval, Boussicault, Bouvel and Silim…
We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the $W$-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset $P$. By proving that the $W$-polynomials of sign-graded posets has the right sign at -1, we are able to prove the Charney-Davis Conjecture for these spheres (whenever they are flag).
There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension $n-d-3$. Moreover, we resolve positively a new special case of the \emph{Generalized Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension $n-d-2$.
In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $Δ$ on the vertex set $V$ with $Δ\ne 2^V$, the deleted join of $Δ$ with its Alexander dual $Δ^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that thes…
Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to sphe…
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.
We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$.
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections an…
RNA molecules are single-stranded analogues of DNA that can fold into various structures which influence their biological function within the cell. RNA structures can be modelled combinatorially in terms of a certain type of graph called an RNA diagram. In this paper we introduce a new poset of RNA diagrams $\mathcal{B}^r_{f,k}$, $r\ge 0$, $k \ge 1$ and $f \ge 3$, which we call the Penner-Waterman poset, and, using results from the theory of multitriangulations, we show that this is a pure poset of rank $k(2f-2k+1)+r-f-1$, whose geometric realization is the join of a simplicial sphere of dimen…
Hypertrees and noncrossing trees are well-established objects in the combinatorics literature, but the hybrid notion of a noncrossing hypertree has received less attention. In this article I investigate the poset of noncrossing hypertrees as an induced subposet of the hypertree poset. Its dual is the face poset of a simplicial complex, one that can be identified with a generalized cluster complex of type $A$. The first main result is that this noncrossing hypertree complex is homeomorphic to a piecewise spherical complex associated with the noncrossing partition lattice and thus it has a natur…
Sharp Bounds on the Independence Number of Simplicial Spheres
Duke University（杜克大学）；University of California, Davis（加州大学戴维斯分校）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd dimensional manifolds without boundary.
Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-vectors. These results allow us to give a complete characterization of the h-vectors of simplicial poset balls up through dimension six.
We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes and the toric $h$-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions \cite{Wal}, allow us to give a complete characterization of the $f$-vectors of arbitrary simplicial triangulations of $S^1 \times S^3, \C P^2,$ $ K3$ surfaces, and $(S^2 \ti…
We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves conjecture on cohomological finiteness conditions for metabelian groups. We find that there is a natural polyhedrality in a crucial special case. In the case of the two dimensional sphere we establish a strong polyhedrality condition for certain open sets which are maximal subject to satisfying the tameness condition that subsets of three or fewer points are contained in open hemispheres. Many examples are included.
Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincaré duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology class in a simplicial poset the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation. As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$ Higher dimensional analogs of discrete normal surfaces …
