Knots and primes : an introduction to arithmetic topology
著者
書誌事項
Knots and primes : an introduction to arithmetic topology
(Universitext)
Springer, 2024
2nd ed
- タイトル別名
-
結び目と素数
Musubime to sosu
大学図書館所蔵 全5件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
English Language edition of 'Musubime to Sosu' Copyright c Springer Japan 2009--T.p. verso
Includes bibliographical references and index
内容説明・目次
内容説明
This book provides a foundation for arithmetic topology, a new branch of mathematics that investigates the analogies between the topology of knots, 3-manifolds, and the arithmetic of number fields. Arithmetic topology is now becoming a powerful guiding principle and driving force to obtain parallel results and new insights between 3-dimensional geometry and number theory.
After an informative introduction to Gauss' work, in which arithmetic topology originated, the text reviews a background from both topology and number theory. The analogy between knots in 3-manifolds and primes in number rings, the founding principle of the subject, is based on the étale topological interpretation of primes and number rings. On the basis of this principle, the text explores systematically intimate analogies and parallel results of various concepts and theories between 3-dimensional topology and number theory. The presentation of these analogies begins at an elementary level, gradually building to advanced theories in later chapters. Many results presented here are new and original.
References are clearly provided if necessary, and many examples and illustrations are included. Some useful problems are also given for future research. All these components make the book useful for graduate students and researchers in number theory, low dimensional topology, and geometry.
This second edition is a corrected and enlarged version of the original one. Misprints and mistakes in the first edition are corrected, references are updated, and some expositions are improved. Because of the remarkable developments in arithmetic topology after the publication of the first edition, the present edition includes two new chapters. One is concerned with idelic class field theory for 3-manifolds and number fields. The other deals with topological and arithmetic Dijkgraaf–Witten theory, which supports a new bridge between arithmetic topology and mathematical physics.
目次
Chapter 1. Introduction.- Chapter 2. Preliminaries - Fundamental Groups and Galois Groups.-Chapter 3. Knots and Primes, 3-Manifolds and Number Rings.- Chapter 4. Linking Numbers and Legendre Symbols.- Chapter 5. Decompositions of Knots and Primes.- Chapter 6. Homology Groups and Ideal Class Groups I – Genus Theory.- Chapter 7. Idelic Class Field Theory for 3-Manifolds and Number Fields.- Chapter 8. Link Groups and Galois Groups with Restricted Ramification.- Chapter 9. Milnor Invariants and Multiple Power Residue Symbols.- Chapter 10. Alexander Modules and Iwasawa Modules.- Chapter 11. Homology Groups and Ideal Class Groups II – Higher Order Genus Theory.- Chapter 12. Homology Groups and Ideal Class Groups III – Asymptotic Formulas.- Chapter 13. Torsions and the Iwasawa Main Conjecture.- Chapter 14. Moduli Spaces of Representations of Knot and Prime Groups.- Chapter 15. Deformations of Hyperbolic Structures and of p-Adic Ordinary Modular Forms.- Chapter 16. Dijkgraaf–Witten Theory for 3-Manifolds and Number Rings.
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