4,475 research outputs found
A Short Simplicial h -Vector and the Upper Bound Theorem
Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42425/1/454-28-3-283_s00454-002-0746-7.pd
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
Topological obstructions for vertex numbers of Minkowski sums
We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i
\ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the
maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of
Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands.
The result is obtained by combining methods from discrete geometry (Gale
transforms) and topological combinatorics (van Kampen--type obstructions) as
developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos.
Construction/example and remarks adde
On r-stacked triangulated manifolds
The notion of r-stackedness for simplicial polytopes was introduced by
McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this
paper, we define the r-stackedness for triangulated homology manifolds and
study their basic properties. In addition, we find a new necessary condition
for face vectors of triangulated manifolds when all the vertex links are
polytopal.Comment: 15 page
Tight triangulations of closed 3-manifolds
It is well known that a triangulation of a closed 2-manifold is tight with
respect to a field of characteristic two if and only if it is neighbourly; and
it is tight with respect to a field of odd characteristic if and only if it is
neighbourly and orientable. No such characterization of tightness was
previously known for higher dimensional manifolds. In this paper, we prove that
a triangulation of a closed 3-manifold is tight with respect to a field of odd
characteristic if and only if it is neighbourly, orientable and stacked. In
consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for
fields of odd characteristic.
Next let be a field of characteristic two. It is known that, in
this case, any neighbourly and stacked triangulation of a closed 3-manifold is
-tight. For triangulated closed 3-manifolds with at most 71
vertices or with first Betti number at most 188, we show that the converse is
true. But the possibility of an -tight non-stacked triangulation on
a larger number of vertices remains open. We prove the following upper bound
theorem on such triangulations. If an -tight triangulation of a
closed 3-manifold has vertices and first Betti number , then
. Equality holds here if and only if all
the vertex links of the triangulation are connected sums of boundary complexes
of icosahedra.Comment: 21 pages, 1 figur
- …