## Galois Groups and Fundamental GroupsEver 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. |

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### Contents

Fundamental groups in topology | 27 |

Riemann surfaces | 65 |

Fundamental groups of algebraic curves | 93 |

Tannakian fundamental groups | 206 |

261 | |

268 | |

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### Common terms and phrases

A-comodule action aﬁnite algebraically closed algebraically closed ﬁeld assume automorphism base change base point branched cover category of ﬁnite closure coefﬁcients commutative comodule compact Riemann surface composite construction Corollary corresponding deﬁned deﬁnition denote discrete valuation ring disjoint union element equivalence Example fact ﬁbre functor ﬁeld extension ﬁnd ﬁnite dimensional ﬁnite étale cover ﬁnite extension ﬁnite Galois ﬁnite group ﬁnite separable ﬁrst ﬁxed follows function ﬁeld fundamental group Galois cover Galois extension Galois group geometric point given Grothendieck group G group scheme holomorphic map induces inﬁnite integral closure integral proper normal inverse limit inverse system isomorphism k-algebra k-linear k-rational k-vector space Lemma locally free sheaf maximal ideal moreover normal subgroup obtain open neighbourhood open subset phism polynomial proﬁnite group projective Proof proper normal curve Proposition Remark representation resp satisﬁes sheaves Spec structure subextension surjective Tannakian Tannakian category Theorem topological space trivial U C X unique