Vol. 13, No. 4, 2020

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Minimal flag triangulations of lower-dimensional manifolds

Christin Bibby, Andrew Odesky, Mengmeng Wang, Shuyang Wang, Ziyi Zhang and Hailun Zheng

Vol. 13 (2020), No. 4, 683–703
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 := f1 5f0 + 16 16β1, 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.

Keywords
flag complexes, triangulations of manifolds, Charney–Davis conjecture
Mathematical Subject Classification
Primary: 05E45, 57Q15
Milestones
Received: 31 March 2020
Revised: 20 July 2020
Accepted: 6 August 2020
Published: 20 November 2020

Communicated by Kenneth S. Berenhaut
Authors
Christin Bibby
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Andrew Odesky
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Mengmeng Wang
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Shuyang Wang
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Ziyi Zhang
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Hailun Zheng
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States