335 research outputs found
A comparison theorem for -vectors of simplicial polytopes
Let denote the number of -dimensional faces of a convex polytope
. Furthermore, let and denote, respectively, the stacked
and the cyclic -dimensional polytopes on vertices. Our main result is
that for every simplicial -polytope , if for some integers and , then for all such that .
For these inequalities are the well-known lower and upper bound
theorems for simplicial polytopes.
The result is implied by a certain ``comparison theorem'' for -vectors,
formulated in Section 4. Among its other consequences is a similar lower bound
theorem for centrally-symmetric simplicial polytopes.Comment: 8 pages. Revised and corrected version. To appear in "Pure and
Applied Mathematics Quarterly
Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres
The aim of this paper is to give a survey of the known results concerning
centrally symmetric polytopes, spheres, and manifolds. We further enumerate
nearly neighborly centrally symmetric spheres and centrally symmetric products
of spheres with dihedral or cyclic symmetry on few vertices, and we present an
infinite series of vertex-transitive nearly neighborly centrally symmetric
3-spheres.Comment: 26 pages, 8 figure
Bier spheres and posets
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.Comment: 15 pages. Revised and slightly extended version; last section
rewritte
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