1,653 research outputs found
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
Minimal flag triangulations of lower-dimensional manifolds
We prove the following results on flag triangulations of 2- and 3-manifolds.
In dimension 2, we prove that the vertex-minimal flag triangulations of
and have 11 and 12 vertices,
respectively. In general, we show that (resp. ) vertices suffice
to obtain a flag triangulation of the connected sum of copies of
(resp. ). In dimension 3, we
describe an algorithm based on the Lutz-Nevo theorem which provides supporting
computational evidence for the following generalization of the Charney-Davis
conjecture: for any flag 3-manifold, ,
where is the number of -dimensional faces and is the first
Betti number over a field. The conjecture is tight in the sense that for any
value of , there exists a flag 3-manifold for which the equality
holds.Comment: 6 figures, 3 tables, 19 pages. Final version with a few typos
correcte
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