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2020 Minimal flag triangulations of lower-dimensional manifolds
Christin Bibby, Andrew Odesky, Mengmeng Wang, Shuyang Wang, Ziyi Zhang, Hailun Zheng
Involve 13(4): 683-703 (2020). DOI: 10.2140/involve.2020.13.683

Abstract

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 P2 and 𝕊1×𝕊1 have 11 and 12 vertices, respectively. In general, we show that 8+3k (resp. 8+4k) vertices suffice to obtain a flag triangulation of the connected sum of k copies of P2 (resp. 𝕊1×𝕊1). 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, γ2:=f15f0+1616β1, where fi is the number of i-dimensional faces and β1 is the first Betti number over a field k. The conjecture is tight in the sense that for any value of β1, there exists a flag 3-manifold for which the equality holds.

Citation

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Christin Bibby. Andrew Odesky. Mengmeng Wang. Shuyang Wang. Ziyi Zhang. Hailun Zheng. "Minimal flag triangulations of lower-dimensional manifolds." Involve 13 (4) 683 - 703, 2020. https://doi.org/10.2140/involve.2020.13.683

Information

Received: 31 March 2020; Revised: 20 July 2020; Accepted: 6 August 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190432
Digital Object Identifier: 10.2140/involve.2020.13.683

Subjects:
Primary: 05E45, 57Q15

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 4 • 2020
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