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Galois Groups and Fundamental Groups

Book Description

Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1 Galois theory of fields
    1. 1.1 Algebraic field extensions
    2. 1.2 Galois extensions
    3. 1.3 Infinite Galois extensions
    4. 1.4 Interlude on category theory
    5. 1.5 Finite étale algebras
  8. 2 Fundamental groups in topology
    1. 2.1 Covers
    2. 2.2 Galois covers
    3. 2.3 The monodromy action
    4. 2.4 The universal cover
    5. 2.5 Locally constant sheaves and their classification
    6. 2.6 Local systems
    7. 2.7 The Riemann–Hilbert correspondence
  9. 3 Riemann surfaces
    1. 3.1 Basic concepts
    2. 3.2 Local structure of holomorphic maps
    3. 3.3 Relation with field theory
    4. 3.4 The absolute Galois group of C(t)
    5. 3.5 An alternate approach: patching Galois covers
    6. 3.6 Topology of Riemann surfaces
  10. 4 Fundamental groups of algebraic curves
    1. 4.1 Background in commutative algebra
    2. 4.2 Curves over an algebraically closed field
    3. 4.3 Affine curves over a general base field
    4. 4.4 Proper normal curves
    5. 4.5 Finite branched covers of normal curves
    6. 4.6 The algebraic fundamental group
    7. 4.7 The outer Galois action
    8. 4.8 Application to the inverse Galois problem
    9. 4.9 A survey of advanced results
  11. 5 Fundamental groups of schemes
    1. 5.1 The vocabulary of schemes
    2. 5.2 Finite étale covers of schemes
    3. 5.3 Galois theory for finite étale covers
    4. 5.4 The algebraic fundamental group in the general case
    5. 5.5 First properties of the fundamental group
    6. 5.6 The homotopy exact sequence and applications
    7. 5.7 Structure theorems for the fundamental group
    8. 5.8 The abelianized fundamental group
  12. 6 Tannakian fundamental groups
    1. 6.1 Affine group schemes and Hopf algebras
    2. 6.2 Categories of comodules
    3. 6.3 Tensor categories and the Tannaka–Krein theorem
    4. 6.4 Second interlude on category theory
    5. 6.5 Neutral Tannakian categories
    6. 6.6 Differential Galois groups
    7. 6.7 Nori’s fundamental group scheme
  13. Bibliography
  14. Index