The flag upper bound theorem for 3- and 5-manifolds

2015/12/22 by Hailun Zheng, Zheng, Hailun · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1512.06958

13 pages

arxiv created 2015/12/22 · arxiv updated 2015/12/23

Abstract

We prove that among all flag 3-manifolds on n vertices, the join of two circles with \lceil(n)/(2) \rceil and \lfloor(n)/(2) \rfloor vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and find all maximizers of the face numbers in this class.

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