Graph Coverings and (Im)primitive Homology: Examples of Low Degree

Abstract

Given a finite covering of graphs $f:Y\to X$, it is not always the case that $H_1(Y;\mathbb{C})$ is spanned by lifts of primitive elements of ${\pi }_1(X)$. 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 $\ge 3$ and order < 1000 satisfying this criterion, all of which are new examples.

Keywords:

• GAP
• homology of finite covers
• p-group
• primitive homology

Acknowledgements

The results presented were obtained as part of the Summer 2020 Columbia Math Summer Undergraduate Research Program at Columbia University. First of all, we would like to express gratitude to the Department of Mathematics at Columbia University and Michael Woodbury for organizing the program. In particular, we extend special thanks to our advisors Nick Salter and Maithreya Sitaraman for their thoughtful guidance throughout the research process. More specifically, we thank Nick Salter for sharing with us both his knowledge and passion for the field and his careful comments on the many revisions this document underwent, and we thank Maithreya Sitaraman for the stimulating discussions and ideas he shared with us. Finally, Nick would like to thank Benson Farb, Sebastian Hensel, Justin Malestein, and Andy Putman for some very helpful correspondence on the topic.

Declaration of Interest

No potential conflict of interest was reported by the author(s).