Minimal flag triangulations of lower-dimensional manifolds

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

doi:10.2140/involve.2020.13.683

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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

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 $R P^{2}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℝ</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math>$ and $S^{1} \times S^{1}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi mathvariant="double-struck">𝕊</mi></mrow><mrow><mn>1</mn></mrow></msup> <mo class="MathClass-bin">×</mo> <msup><mrow><mi mathvariant="double-struck">𝕊</mi></mrow><mrow><mn>1</mn></mrow></msup></math>$ have 11 and 12 vertices, respectively. In general, we show that $8 + 3 k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn> <mo class="MathClass-bin">+</mo> <mn>3</mn><mi>k</mi></math>$ (resp. $8 + 4 k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn> <mo class="MathClass-bin">+</mo> <mn>4</mn><mi>k</mi></math>$ ) vertices suffice to obtain a flag triangulation of the connected sum of $k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math>$ copies of $R P^{2}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℝ</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math>$ (resp. $S^{1} \times S^{1}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi mathvariant="double-struck">𝕊</mi></mrow><mrow><mn>1</mn></mrow></msup> <mo class="MathClass-bin">×</mo> <msup><mrow><mi mathvariant="double-struck">𝕊</mi></mrow><mrow><mn>1</mn></mrow></msup></math>$ ). 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, $γ_{2} : = f_{1} - 5 f_{0} + 16 \geq 16 β_{1}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msub> <mo class="MathClass-punc">:</mo><mo class="MathClass-rel">=</mo> <msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub> <mo class="MathClass-bin">−</mo> <mn>5</mn><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub> <mo class="MathClass-bin">+</mo> <mn>1</mn><mn>6</mn> <mo class="MathClass-rel">≥</mo> <mn>1</mn><mn>6</mn><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math>$ , where $f_{i}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math>$ is the number of $i$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math>$ -dimensional faces and $β_{1}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math>$ is the first Betti number over a field $k$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="bold-italic"><mi>k</mi></mstyle></math>$ . The conjecture is tight in the sense that for any value of $β_{1}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math>$ , 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

Vol. 13, No. 4, 2020