## Algebraic TopologyBased on lectures to advanced undergraduate and first-year graduate students, this is a thorough, sophisticated and modern treatment of elementary algebraic topology, essentially from a homotopy theoretic viewpoint. The presentation of the homotopy theory and the account of duality in homology manifolds make the text ideal for a course on either homotopy or homology theory. The idea of algebraic topology is to translate problems in topology into problems in algebra with the hope that they have a better chance of solution. The translation process is usually carried out by means of the homology or homotopy groups of a topological space. Much of the book is therefore concerned with the construction of these algebraic invariants, and with applications to topological problems, such as the classification of surfaces and duality theorems for manifolds. Other important topics covered are homotopy theory, CW-complexes and the co-homology groups associated with a general Ω-spectrum. Dr. Maunder has provided many examples and exercises as an aid, and the notes and references at the end of each chapter trace the historical development of the subject and also point the way to more advanced results. "Throughout the text the style of writing is first class. The author has given much attention to detail, yet ensures that the reader knows where he is going. An excellent book." — Bulletin of the Institute of Mathematics and Its Applications. |

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### Contents

Introduction | ix |

Homotopy and Simplicial Complexes | 23 |

The Fundamental Group | 63 |

Homology Theory | 104 |

Cohomology and Duality Theorems | 158 |

General Homotopy Theory | 200 |

Homotopy Groups and CWComplexes | 257 |

Chapter 8 | 311 |

361 | |

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### Common terms and phrases

algebraic base point based map block dissection calculation chain complex chain homotopy chain map Chapter Cn(K coefficients commutative diagram composite consider constant map construct contained continuous map Corollary corresponding CW-complex Deduce defined Definition Duality Theorem element exact sequence Example Exercise exists finite number follows free abelian group fundamental group Given group G Hausdorff Hence Hn(C Hn(K Hn(Mn Hn(X homology groups homology n-manifold homomorphism homotopy classes homotopy equivalence identification map identified identity isomorphism identity map inclusion map induces integer interior inverse isomorphism Lemma linear manifolds map f map of pairs Moreover n-manifold with boundary n-simple open set path component path-connected polyhedra polyhedron properties Proposition prove result Section Similarly simplex Simplicial Approximation Theorem simplicial complex simplicial map singular n-simplex space obtained subcomplex subgroup subset subspace suppose topological space topology triangulation vector space vertex vertices