Galois Groups and Fundamental Groups on Riemann Surfaces


Bachelor Thesis, 2018
40 Pages, Grade: 1,0

Abstract or Introduction

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.

Details

Title
Galois Groups and Fundamental Groups on Riemann Surfaces
College
Free University of Berlin  (Mathematik)
Grade
1,0
Author
Year
2018
Pages
40
Catalog Number
V445009
ISBN (eBook)
9783668818965
ISBN (Book)
9783668818972
Language
English
Tags
Kategorien, Categories, Galois, Profinite, Completion, Fundamental, Group, Coverings, Überdeckungen, Überdeckung, Profinit, Vervollständigung, Gruppe, Gruppentheorie, Riemann, Riemannsche, Fläche, Meromorphic, Functions, Meromorph, Universal, Finite étale, Algebra, Hawaiian earring, Hawaiischer Ohrring, Fundamentalgruppe, Topologie, Topology, Szamuely, Geometrie, algebraische, algebraic, Geometry, cone
Quote paper
Matthias Himmelmann (Author), 2018, Galois Groups and Fundamental Groups on Riemann Surfaces, Munich, GRIN Verlag, https://www.grin.com/document/445009

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