Quandles and Topological Pairs: Symmetry, Knots, and Cohomology
Quandles and Topological Pairs: Symmetry, Knots, and Cohomology
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Quandles 和拓扑对：对称性、结和上同调
DOI：
10.1007/978-981-10-6793-8
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发表时间：
2017
影响因子：
0.5
通讯作者：
Takefumi Nosaka
中科院分区：
文献类型：
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作者：
Takefumi Nosaka
这本书调查quandle理论，从基本动机开始，并通过介绍最近的发展和我自己的研究结果。拓扑应用和相关的主题，提出了整个。虽然拓扑方面的重点，我也想代表如何quandle理论在数学评价。因此，这本书是一个在困境理论速成班。前四章包含了我认为是quandles的基础知识，旨在用quandles来表达拓扑对象。接下来的章节着重从同伦理论和群上同调的角度对这些对象进行了详细的研究。这些章节反映了我自己的研究兴趣和应用。在大多数情况下，作为阅读这本书的先决条件，读者需要代数（群，环，模和同调代数）和代数拓扑（C1-流形，基本群，覆盖空间，CW-复形和（余）-同调）的元素。然而，关于低维拓扑，我在附录A中给出了一个没有证明的基本符号和事实的列表;读者可以简单地相信这些事实（关于细节，见本书末尾的参考文献）。我给出了一些练习，其中一些暗示了任意的缩写，以缩短长的证明。所以我只引用了参考资料作为答案。我认为细节在这本书中不是那么重要，或者细节涉及繁琐的计算。最后，我衷心感谢秋田俊之、J. Scott Carter、Katsumi石川、刘烨、Hirofumi Niibo、Józef H.感谢Przytycki、Masahico Saito、Sumire Sawada、Masayoshi Tanno和Seung Yeop Yang仔细阅读了本书的早期草稿，并给了我一些他们的详细评论。我也感谢裁判员提出的有益意见。此外，我非常感谢大泽武夫教授，我有机会与他一起工作。此外，这本书是我在东京大学和北海道大学为两门强化课程写的笔记的产物。我也感谢出席的学生和教授。川澄奈也和坂斋拓也
This book surveys quandle theory, starting from basic motivation and passing on to introduce recent developments and the results of my own research. Topological applications and related topics are presented throughout. While the topological aspects are the focus, I also would like to represent how quandle theory is evaluated in mathematics. Thus, this book is a crash course in quandle theory. The first four chapters contain what I consider to be the basics of quandles and aim to express topological objects in terms of quandles. The remaining chapters focus on studying the objects in detail from the viewpoints of homotopy theory and group cohomology. These chapters reflect my own research interests and applications. For the most part, as prerequisites for reading this book, the reader needs elements of algebra (groups, rings, modules, and homological algebra) and of algebraic topology (C 1-manifold, fundamental group, covering spaces, CW-complexes, and (co)-homology). However, concerning low-dimensional topology, I give a list of elementary notation and facts in Appendix A without proof; the reader can simply take these facts on faith (for the details, see the references at the end of this book).I give a number of exercises, some of which imply arbitrary abbreviations to cut a long proof short. So I only cited references for the answers. I think that the details are not so essential in this book, or the details involve tedious computations. In closing, I sincerely express my thanks to Toshiyuki Akita, J. Scott Carter, Katsumi Ishikawa, Ye Liu, Hirofumi Niibo, Józef H. Przytycki, Masahico Saito, Sumire Sawada, Masayoshi Tanno, and Seung Yeop Yang for careful reading earlier drafts of this book and for giving me a number of their detailed comments. I also express my appreciation to the referees for his or her useful comments. Furthermore, I gratefully thank Prof. Takeo Ohsawa with whom I have had the opportunity of working on this publication. Moreover, this book grew out of notes I wrote for two intensive courses at The University of Tokyo, and the Hokkaido University. I also thank the attending students and Profs. Nariya Kawazumi and Takuya Sakasai.