Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres.

IF 0.6 3区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Discrete & Computational Geometry Pub Date : 2025-01-01 Epub Date: 2025-05-13 DOI:10.1007/s00454-025-00733-6
Cesar Ceballos, Joseph Doolittle
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引用次数: 0

Abstract

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani (Aequationes Math 34(2-3):287-297, 1987, 10.1007/BF01830678), via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai (2009, https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/). In this paper, we show that Kalai's conjecture holds in the particular case of Knutson and Miller's spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.

子词复合体与卡莱关于球的重构猜想。
多面体理论中的一个著名定理指出,一个简单多面体的组合类型完全由它的面脊图决定。Blind和Mani (aequesmath34 (2-3):287-297, 1987, 10.1007/BF01830678)利用同调理论的拓扑工具,通过非建设性证明证明了这个著名的结果。不久之后,卡莱给出了一个优雅的建设性的证明。Blind和Mani在他们的原始论文中提出,他们的结果是否可以推广到简单球体,Kalai (2009, https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/)对他们的问题做出了肯定的回答。在本文中,我们证明Kalai的猜想在Knutson和Miller的球形子词复合体的特殊情况下成立。这个家族的简单球体出现在考克斯特群的背景下,并被推测为多面体。相反,并非所有流形都是可重构的。我们给出两个明确的例子,即环面和射影平面。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Discrete & Computational Geometry
Discrete & Computational Geometry 数学-计算机:理论方法
CiteScore
1.80
自引率
12.50%
发文量
99
审稿时长
6-12 weeks
期刊介绍: Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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