## Algebraic Topology Via Differential GeometryIn this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. |

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### Contents

ALGEBRAIC PRELIMINARIES | 1 |

DIFFERENTIAL FORMS ON AN OPEN SUBSET OF Rⁿ | 30 |

DIFFERENTIABLE MANIFOLDS | 70 |

DE RHAM COHOMOLOGY OF DIFFERENTIABLE MANIFOLDS | 121 |

COMPUTING COHOMOLOGY | 161 |

POINCARE DUALITY LEFSCHETZ THEOREM | 214 |

### Common terms and phrases

basis bilinear chart cohomology compact subset compact supports complex connexion constant map continuous map Corollary definition degree denoted diffeomorphism differentiable manifold differential form direct sum E B F element exact sequence exists a unique exterior algebra ferentiable manifold graded algebra Grassmann manifolds Gysin homomorphism H CRP hence homomorphism homotopy inclusion induces integer inverse isomorphism Jacobian matrix k-form Lemma linear map manifold of dimension manifold with boundary map f maximal atlas morphism n-dimensional orientable neighbourhood non-zero notations obtain open ball open subset oriented differentiable manifold partition of unity Poincare Poincare duality Proof Remark resp restriction Rham cohomology satisfies shows submanifold Supp Theorem topological manifold vector bundle vector space volume form x e H zero