球面与戴维斯流形之间的简易映射,具有正简易体积

Francesco Milizia
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引用次数: 0

摘要

我们研究由戴维斯反射群技巧得到的流形的单纯体积,目的是描述那些具有正单纯体积的流形的特征。特别是,我们将重点放在检验该类流形中具有非零欧拉特征的流形是否具有正单纯容积(格罗莫夫曾问这是否在一般情况下对非球面流形成立)。这就引出了一个关于球面三角剖分的组合问题:我们在三角剖分集合上定义了一个偏序--关系是两个三角剖分之间存在一个非零度的简并映射--问题是找到特定子集的最小元素。我们明确地解决了二维球体的三角形问题,然后借助计算机搜索对三维问题进行了深入分析。此外,我们还提出了这一问题与图最小值理论的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Simplicial maps between spheres and Davis' manifolds with positive simplicial volume
We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.
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