## 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

algebra Apply associated assume atlas ball bases basis bilinear boundary bundle canonical Chapter chart clearly closed cº-map cohomology commutative compact compact supports complex consider consists construction continuous map Corollary course covering defined definition denoted determinant diffeomorphism differentiable manifold differential form dimension direct element equal exact sequence example exists fact Figure Finally finite finite-dimensional given gives graded hence homomorphism inclusion induced integer introduction isomorphism Lemma linear map matrix module n-dimensional neighbourhood notations notion obtain open subset oriented differentiable particular possible projection Proof properties prove Remark resp respectively restriction result satisfies shows Similarly situation structure Supp Theorem topological manifold trivial unique vector space volume form write zero