This thesis deals with the correlation of the fundamental group and the Galois group, using their corresponding entities of covering spaces and field extensions. First it is viewed in the general setting of categories, using the language of Galois categories. It is shown that the categories of the finite étale algebras and the category of covering spaces are correlated, which gives the fact that the profinite completion of the fundamental group and the absolute Galois group are similar. More specifically, on Riemann surfaces it is shown that there exists an anti-equivalence of categories between the finite field extensions of the meromorphic functions of a compact, connected Riemann Surface X and the category of branched coverings of X. A more explicit theorem, that provides an isomorphism between a specific Galois Group and the profinite Completion of the Fundamental Group of a pointed X, gives more insight on the behaviour of these two groups.
Inhaltsverzeichnis (Table of Contents)
- Algebraic Foundations
- Category Theory
- Profinite Groups
- Finite Field Extensions
- The Fundamental Group
- Covering Spaces
- Galois Categories
- Definition
- Infinite Galois Theory
- Finite Étale Algebras
- Covering Spaces
- Universal Cover
- Coverings with marked points
- The profinite completion of the Fundamental Group
- Riemann Surfaces
- Riemann Surfaces
- Meromorphic Functions
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The primary aim of this thesis is to establish a connection between the fundamental group and the Galois group in the context of covering spaces. This is achieved by examining the category of finite covering spaces and the category of specific k-algebras over a field k. Furthermore, the thesis delves into the relationship between the separable closure of a field and the universal cover of a topological space. This exploration aims to expand the Fundamental Theorem of Galois Theory to encompass infinite Galois extensions.
- Connection between fundamental groups and Galois groups
- Category theory applications in the context of covering spaces and k-algebras
- Comparison of separable closures and universal covers
- Extension of the Fundamental Theorem of Galois Theory to infinite Galois extensions
- Explicit demonstration of the relationship between fundamental groups and Galois groups in Riemann surfaces
Zusammenfassung der Kapitel (Chapter Summaries)
Chapter 1 lays the foundation by reviewing key concepts from category theory, algebra, and topology, including definitions of categories, functors, morphisms, and the fundamental group. Chapter 2 delves into the definition and properties of Galois categories, including infinite Galois theory and finite étale algebras. Chapter 3 focuses on the concept of covering spaces, examining the universal cover, coverings with marked points, and the profinite completion of the fundamental group.
Schlüsselwörter (Keywords)
The thesis explores concepts such as category theory, fundamental groups, Galois groups, covering spaces, universal covers, profinite completion, finite étale algebras, Riemann surfaces, meromorphic functions, and infinite Galois theory.
- Quote paper
- Matthias Himmelmann (Author), 2018, Galois Groups and Fundamental Groups on Riemann Surfaces, Munich, GRIN Verlag, https://www.grin.com/document/445009
