论文标题
通过大量完成区分4维几何形状
Distinguishing 4-dimensional geometries via profinite completions
论文作者
论文摘要
众所周知，从瑟斯顿（Thurston）的意义上讲，有19种的几种几何形状。我们可以问，几何信息在多大程度上通过封闭的平滑几何4个manifold的基本组的完整组完成。在本文中，我们表明，从瑟斯顿的意义上讲，可以通过其基本组的完整完成，除了几何形状为$ \ mathbb {h}^{4} {4} $，$ \ \ \ \ \ \ mathb {h}^{2}^{2}^{c { $ \ mathbb {h}^2 \ times \ mathbb {h}^2 $。此外，尽管并非每个平滑的4个曼物都可以从瑟斯顿的意义上接受一个几何形状，但一些具有塞弗特纤维结构的四维流形的几何形状确实是几何的。对于可闭合的可定向的Seifert Fibred 4-manifold $ M $，我们表明，是否可以通过其基本组的仔细完成$ M $几何来检测几何。
It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ or $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group.
