nach oben

2012 | Buch

Kapitel lesen Erstes Kapitel lesen

Knots and Primes

An Introduction to Arithmetic Topology

verfasst von: Masanori Morishita

Verlag: Springer London

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory. Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Common aspects of both knot theory and number theory, for instance knots in three manifolds versus primes in a number field, are compared throughout the book. These comparisons begin at an elementary level, slowly building up to advanced theories in later chapters. Definitions are carefully formulated and proofs are largely self-contained. When necessary, background information is provided and theory is accompanied with a number of useful examples and illustrations, making this a useful text for both undergraduates and graduates in the field of knot theory, number theory and geometry.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Starting with the work of Gauss on quadratic residues and linking numbers, we review some histories of knot theory and number theory that branched out after Gauss. In particular, we trace the string of thoughts on geometrization of number theory which led to the theme of this book, arithmetic topology—a new branch of mathematics bridging between knot theory and number theory. An outline of this book is also included.

Masanori Morishita

Chapter 2. Preliminaries—Fundamental Groups and Galois Groups

Abstract

In this chapter we recollect the preliminary materials from topology and number theory. In particular, we give a summary about fundamental groups and Galois theory for topological spaces and arithmetic rings, together with the basic concepts and examples in 3-dimensional topology and number fields. We also review class field theory as arithmetic duality theorems in Galois, étale cohomology groups.

Masanori Morishita

Chapter 3. Knots and Primes, 3-Manifolds and Number Rings

Abstract

In this chapter we explain the basic analogies between knots and primes, 3-manifolds and number rings and present a dictionary of these analogies, which will be fundamental and used throughout subsequent chapters.

Masanori Morishita

Chapter 4. Linking Numbers and Legendre Symbols

Abstract

In this chapter we discuss the close analogy between the linking number and the Legendre symbol, based on the analogies between knots and primes in Chap. 3. An analogy between Gauss’ linking integral and Gauss sum is also pointed out.

Masanori Morishita

Chapter 5. Decompositions of Knots and Primes

Abstract

In this chapter we review Hilbert theory which deals with, in a group-theoretic manner, the decomposition of a prime in a finite Galois extension of number fields. Based on the analogies in Chap. 3, we give a topological analogue of Hilbert theory for a finite Galois covering of 3-manifolds.

Masanori Morishita

Chapter 6. Homology Groups and Ideal Class Groups I—Genus Theory

Abstract

In this chapter we review Gauss’ genus theory from the link-theoretic point of view. We shall see that the notion of genera is defined by using the idea analogous to the linking number. We also present, vice versa, a topological analogue of genus theory.

Masanori Morishita

Chapter 7. Link Groups and Galois Groups with Restricted Ramification

Abstract

In this chapter we discuss the analogy between Galois groups with restricted ramification and link groups. In particular, we shall see the close analogy between Milnor’s theorem on the structure of a link group and a theorem by H. Koch on the structure of a pro-l Galois group over the rational number field with restricted ramification.

Masanori Morishita

Chapter 8. Milnor Invariants and Multiple Residue Symbols

Abstract

In this chapter we take up the Milnor invariants, higher order linking numbers, of a link in the 3-sphere, together with the Fox free differential calculus as a basic tool. By the analogy between a link group and a Galois group with restricted ramification in Chap. 7, we introduce arithmetic analogues of the Milnor invariants for prime numbers by using the pro-l Fox differential calculus, and show that they may be regarded as multiple generalization of the power residue symbol and the Rédei triple symbol.

Masanori Morishita

Chapter 9. Alexander Modules and Iwasawa Modules

Abstract

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

Masanori Morishita

Chapter 10. Homology Groups and Ideal Class Groups II—Higher Order Genus Theory

Abstract

In this chapter we first study the 2-part of the 1st homology group of a double covering of the 3-sphere ramified over a link, introducing the higher order linking matrices which are defined by using the Milnor numbers of the link in Chap. 8. Imitating the method for a link, we study the 2-part of the narrow ideal class group of a quadratic extension of the rationals, using the arithmetic Milnor numbers introduced in Chap. 8. Our theorem may be regarded as a higher order generalization of Gauss’ and Rédei’s theorems on the 2-rank and 4-rank of the ideal class group.

Masanori Morishita

Chapter 11. Homology Groups and Ideal Class Groups III—Asymptotic Formulas

Abstract

In this chapter we discuss close parallels between Alexander-Fox theory and Iwasawa theory such as the Alexander polynomial and the Iwasawa polynomial, based on the analogy between the infinite cyclic covering of a knot exterior and the cyclotomic ℤ_p-extension of a number field. In particular, we give asymptotic formulas on the orders of the 1st homology groups (p-ideal class groups) of cyclic ramified coverings (extensions).

Masanori Morishita

Chapter 12. Torsions and the Iwasawa Main Conjecture

Abstract

In this chapter we take up the Iwasawa main conjecture which asserts that the Iwasawa polynomial coincides essentially with the Kubota–Leopoldt p-adic analytic zeta function. According to the analogy between the Iwasawa polynomial and the Alexander polynomial in Chap. 11, we discuss geometric analogues of the Iwasawa main conjecture, namely, some relations between the Reidemeister–Milnor torsion and the Lefschetz or spectral zeta function.

Masanori Morishita

Chapter 13. Moduli Spaces of Representations of Knot and Prime Groups

Abstract

In this chapter, in view of the analogy between a knot group and a prime group, we discuss some analogies between the moduli spaces of 1-dimensional representations of knot and prime groups, and the associated higher Alexander and Iwasawa invariants.

Masanori Morishita

Chapter 14. Deformations of Hyperbolic Structures and p-Adic Ordinary Modular Forms

Abstract

In this final chapter, as an extension of the analogies in Chap. 13 to 2-dimensional representations, we discuss intriguing analogies between the family of holonomy representations associated to deformations of hyperbolic structures (due to W. Thurston) and the family of Galois representations associated to deformations of p-adic ordinary modular forms (due to H. Hida and B. Mazur).

Masanori Morishita

Backmatter

Titel: Knots and Primes
verfasst von: Masanori Morishita
Verlag: Springer London
Electronic ISBN: 978-1-4471-2158-9
Print ISBN: 978-1-4471-2157-2
DOI: https://doi.org/10.1007/978-1-4471-2158-9