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.
Table of Contents
1 Algebraic Foundations
1.1 Category Theory
1.2 Profinite Groups
1.3 Finite Field Extensions
1.4 The Fundamental Group
1.5 Covering Spaces
2 Galois Categories
2.1 Definition
2.2 Infinite Galois Theory
2.3 Finite Etale Algebras
3 Covering Spaces
3.1 Universal Cover
3.2 Coverings with marked points
3.3 The profinite completion of the Fundamental Group
4 Riemann Surfaces
4.1 Riemann Surfaces
4.2 Meromorphic Functions
Research Objectives and Core Topics
This thesis aims to establish a formal correlation between the fundamental group of a topological space and the Galois group of a field extension, utilizing the language of category theory to analyze finite covering spaces and finite étale algebras. It seeks to prove isomorphisms between these algebraic and topological structures, specifically within the context of Riemann surfaces and their branched coverings.
- Category Theory and Galois Categories
- Infinite Galois Theory and Profinite Groups
- Fundamental Groups and Covering Space Theory
- Finite Étale Algebras
- Complex Geometry and Riemann Surfaces
Excerpt from the Book
1.1 Category Theory
Definition 1.1.1 A category C consists of a class of objects ob(C) and a class Hom(C) of morphisms between those objects. Given two objects A and B, we write HomC(A, B) for the set of morphisms A → B. We require:
1. for φ ∈ HomC(A, B) and ψ ∈ HomC(B, D) there is ψ ◦ φ ∈ HomC(A, D) and we call it composition. For this, we require associativity.
2. ∀A ∈ ob(C) ∃ idA ∈ HomC(A, A), the identity morphism, that fulfils φ ◦ idA = φ = idB ◦ φ for any φ ∈ HomC(A, B).
In the following, we will only deal with small categories, meaning that the objects of this category form a set.
Definition 1.1.2 For a category C, the opposite category Cop is defined by ob(Cop) = ob(C) and reversing the morphisms, so we have HomC(A, B) = HomCop(B, A) for any A, B ∈ ob(C).
Definition 1.1.3 In a category C, an object Z is called final, if for each object B there is a unique morphism B → Z. Conversely, an object is called initial, if for each object B there is a unique morphism A → B.
Summary of Chapters
1 Algebraic Foundations: Reviews fundamental concepts from category theory, profinite groups, field theory, and topology required for the subsequent analysis.
2 Galois Categories: Introduces the axioms of a Galois category and demonstrates that finite étale algebras over a field form such a category.
3 Covering Spaces: Explores the category of finite coverings, identifies it as a Galois category, and examines the profinite completion of the fundamental group.
4 Riemann Surfaces: Provides a concrete application, showing the correspondence between branched coverings and finite field extensions of meromorphic functions on compact Riemann surfaces.
Keywords
Galois Group, Fundamental Group, Riemann Surface, Category Theory, Covering Space, Profinite Group, Field Extension, Étale Algebra, Galois Category, Meromorphic Function, Branched Covering, Universal Cover, Topology, Algebra, Isomorphism.
Frequently Asked Questions
What is the fundamental objective of this research?
The work aims to rigorously prove and formalize the conceptual bridge between the fundamental group of a topological space and the Galois group of a field extension, showing they are different sides of the same mathematical phenomena.
What are the primary fields of study involved?
The thesis intersects Algebraic Geometry, Algebraic Topology, and Field Theory, specifically focusing on the intersection of Riemann surfaces, Galois extensions, and category theory.
What is the core research question?
The question addresses how one can establish an explicit isomorphism between the Galois group of specific field extensions and the profinite completion of the fundamental group of related punctured topological spaces.
Which mathematical framework is used to conduct this study?
The author employs the framework of Galois categories, using fundamental functors to translate topological covering data into algebraic categories of sets.
What is the significance of the "last chapter" in the thesis?
The final chapter serves as an explicit case study using Riemann surfaces to illustrate how the abstract correlations derived in earlier chapters manifest in concrete geometric and complex-analytic structures.
How would you summarize the work in a few keywords?
Key terms include Galois groups, fundamental groups, covering spaces, étale algebras, and Riemann surfaces.
How is the Hawaiian Earring Group used in the text?
The Hawaiian Earring group is used as a counter-example to demonstrate spaces that are locally path-connected but fail to be semi-locally simply-connected, illustrating why certain topological conditions are necessary for the existence of universal covers.
What does the thesis conclude regarding the connection between fundamental groups and Galois groups?
It concludes that under certain conditions, specifically regarding the profinite completion of the fundamental group, a direct and natural correspondence exists between these groups, effectively showing they share the same algebraic structure.
- Citar trabajo
- Matthias Himmelmann (Autor), 2018, Galois Groups and Fundamental Groups on Riemann Surfaces, Múnich, GRIN Verlag, https://www.grin.com/document/445009
