## Manifolds, Sheaves, and CohomologyThis book explains techniques that are essential in almost all branches of modern geometry such as algebraic geometry, complex geometry, or non-archimedian geometry. It uses the most accessible case, real and complex manifolds, as a model. The author especially emphasizes the difference between local and global questions. Cohomology theory of sheaves is introduced and its usage is illustrated by many examples. |

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### Contents

1 | |

2 Algebraic Topological Preliminaries | 21 |

3 Sheaves | 41 |

4 Manifolds | 69 |

5 Linearization of Manifolds | 91 |

6 Lie Groups | 123 |

7 Torsors and Nonabelian Čech Cohomology | 138 |

8 Bundles | 153 |

11 Cohomology of Constant Sheaves | 233 |

Basic Topology | 245 |

The Language of Categories | 270 |

Basic Algebra | 291 |

Homological Algebra | 317 |

Local Analysis | 331 |

References | 341 |

343 | |

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Â X open A-linear map abelian groups algebra assume basis bijective C*-premanifold called chart cohomology colimits commutative ring composition connected components continuous map Corollary covering map define denoted diagram element endowed equivalent étalé spaces exact sequence Example exists a unique exists an open f WX fiber bundle finite locally free functions Hausdorff space Hence homomorphism homotopy induces injective isomorphism K-algebras K-injective K-linear Lemma Let F Let G Lie group locally constant locally finite manifold map f modules morphism of premanifolds morphism of sheaves notion numbers objects open covering open neighborhood open subsets open subspace OX-modules particular path connected phism presheaf Problem Proof quotient R-modules R-ringed real analytic ringed space second countable sheafification Show space and let Springer structure sheaf subgroup submanifold submersion subpremanifold surjective Theorem topological space typical fiber vector bundles yields