Abstract

Given a finite covering of graphs f:Y→X$f:Y\to X$$f:Y\to X$, it is not always the case that H1(Y;C)$H_1(Y;\mathbb{C})$$H_1(Y;\mathbb{C})$ is spanned by lifts of primitive elements of π1(X)${\pi }_1(X)$${\pi }_1(X)$. In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite p-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite p-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite p-groups of rank ≥3$\ge 3$$\ge 3$ and order < 1000 satisfying this criterion, all of which are new examples.

摘要

给定图的有限覆盖F:是→ X$F:是\to X$$f:Y\to X$，情况并不总是这样H1(是; C ）$H_1（是;\mathbb{C}）$$H_1(Y;\mathbb{C})$是由原始元素的升力跨越的π1( X）${\pi }_1（X）$${\pi }_1(X)$。在本文中，我们研究的图并非如此，我们在这里给出了具有此属性的最简单的已知非平凡覆盖示例，覆盖度小至 128。我们的第一步是将注意力集中在特殊类别图表涵盖了甲板组是有限p组的情况。对于此类覆盖物，存在用于识别存在具有该属性的覆盖物的套牌组的表示理论标准。我们提出了一种算法来确定有限p群是否满足此标准，该算法仅使用该群的字符表。最后，我们提供了所有有限p组秩的完整普查≥3 _$\ge 3$$\ge 3$并且 order < 1000 满足这个标准，都是新的例子。