Computing the homology of universal covers via effective homology and discrete vector fields

Miguel Angel Marco-Buzunariz, Ana Romero
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Abstract

Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted cartesian product. As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields. Some examples showing our implementation of these constructions in both \sagemath\ and \kenzo\ are shown, together with an approach to compute the homology of the universal cover when the group is abelian even in some cases where there is no effective homology, using the twisted homology of the space.
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通过有效同源性和离散向量场计算普遍盖的同源性
通过有效的同调技术,我们可以计算拓扑空间广族的同调群。通过怀特海塔方法,这也可以用来计算高等同调群。然而,其中一些技术(特别是怀特海塔)依赖于起始空间是简单连接的假设。在某些应用中,这个问题可以通过用通用盖来代替空间来规避,通用盖是一个简单连接的空间,它共享初始空间的高次同调群。在本文中,我们正式提出了通用盖的简单构造,并将其表示为扭曲的笛卡尔乘积。正如我们通过一些例子所展示的,具有有效同调的空间的普遍盖在一般情况下并不一定具有有效同调。我们展示了能确保它的两个独立充分条件:一个是基于基本群的无效性质,另一个是基于离散向量场。我们举了一些例子来说明这些构造在《sagemath\ 》和《\kenzo\》中的实现,同时还展示了一种方法,当群是无性的甚至在某些情况下没有有效同调时,利用空间的扭曲同调来计算普遍盖的同调。
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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