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Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres
Abstract
We study the twelve-vertex flag triangulation T12 of S3 and the Davis $4$-manifold M(T12) associated with it. We prove that T12 is, up to isomorphism, the unique flag triangulation of S3 with twelve vertices in which every edge is contained in a square; this proves Milizia's Conjecture~6.31. We also prove a general criterion for the tangent Euler class of a Davis $4$-manifold to be bounded, and we verify it for T12 by means of an explicit symmetric form. Since the Euler class takes the value $256$ on the fundamental class of M(T12), the duality between bounded cohomology and simplicial volume gives |M(T12)|>0. This answers Milizia's Question~1.7.
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BibTeX
@misc{lai2026positivity,
title = {{Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres}},
author = {Xiaojun Lai and Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34743},
eprint = {2610.34743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34743},
}
RIS
TY - UNPB
TI - Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres
AU - Xiaojun Lai
AU - Xueyuan Wan
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34743
UR - https://arxiv.org/abs/2610.34743
AB - We study the twelve-vertex flag triangulation T_12 of S^3 and the Davis $4$-manifold M(T_12) associated with it. We prove that T_12 is, up to isomorphism, the unique flag triangulation of S^3 with twelve vertices in which every edge is contained in a square; this proves Milizia's Conjecture~6.31. We also prove a general criterion for the tangent Euler class of a Davis $4$-manifold to be bounded, and we verify it for T_12 by means of an explicit symmetric form. Since the Euler class takes the value $256$ on the fundamental class of M(T_12), the duality between bounded cohomology and simplicial volume gives |M(T_12)|>0. This answers Milizia's Question~1.7.
ER -