## A Basic Course in Algebraic TopologyThe main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. It is in some sense a sequel to the author's previous book in this Springer-Verlag series entitled Algebraic Topology: An Introduction. This earlier book is definitely not a logical prerequisite for the present volume. However, it would certainly be advantageous for a prospective reader to have an acquaintance with some of the topics treated in that earlier volume, such as 2-dimensional manifolds and the funda mental group. Singular homology and cohomology theory has been the subject of a number of textbooks in the last couple of decades, so the basic outline of the theory is fairly well established. Therefore, from the point of view of the mathematics involved, there can be little that is new or original in a book such as this. On the other hand, there is still room for a great deal of variety and originality in the details of the exposition. In this volume the author has tried to give a straightforward treatment of the subject matter, stripped of all unnecessary definitions, terminology, and technical machinery. He has also tried, wherever feasible, to emphasize the geometric motivation behind the various concepts. |

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### Contents

4 Examples of Compact Connected 2Manifolds | 5 |

K Triangulations of Compact Surfaces | 14 |

The Euler Characteristic of a Surface | 26 |

The Fundamental Group | 35 |

4 The Effect of a Continuous Mapping on the Fundamental Group | 42 |

The Brouwer FixedPoint Theorem in Dimension 2 | 51 |

CHAPTER | 60 |

4 Free Products of Groups | 71 |

Homology of CWCompIexes | 225 |

4 The Homology Groups of a CWComplex | 232 |

5 Incidence Numbers and Orientations of Cells | 238 |

7 Determination of Incidence Numbers | 244 |

CHAPTER X | 254 |

3 Definition and Basic Properties of Homology with | 262 |

6 The Universal Coefficient Theorem | 269 |

CHAPTER XI | 279 |

6 The Presentation of Groups by Generators and Relations | 78 |

Seifert and Van Kampen Theorem on the Fundamental Group | 86 |

4 Second Application of Theorem 2 1 | 95 |

6 Application to Knot Theory | 103 |

Covering Spaces | 117 |

10 The Existence Theorem for Covering Spaces | 140 |

CHAPTER VI | 147 |

CHAPTER VII | 158 |

4 The Homotopy Property of the Induced Homomorphisms | 166 |

The Main Properties of Relative Homology Groups | 173 |

CHAPTER VIII | 186 |

3 Homology of Finite Graphs | 192 |

4 Homology of Compact Surfaces | 201 |

5 The MayerVietoris Exact Sequence | 207 |

7 The Relation between the Fundamental Group | 217 |

Proof of the EilenbergZilber Theorem | 286 |

Formulas for the Homology Groups of Product Spaces | 300 |

CHAPTER XII | 305 |

5 Geometric Interpretation of Cochains Cocycles etc | 316 |

CHAPTER XIII | 323 |

10 Remarks on the Coefficients for the Various Products | 343 |

Duality Theorems for the Homology of Manifolds | 350 |

3 Cohomology with Compact Supports | 358 |

5 Applications of the Poincare Duality Theorem | 365 |

7 Duality Theorems for Manifolds with Boundary | 375 |

394 | |

H The Hopf Invariant | 402 |

2 Differentiable Singular Chains | 408 |

419 | |

### Common terms and phrases

3-space algebraic topology arcwise connected assert Assume automorphism boundary operator chain complexes chain homotopy chain map Chapter choose circle closed path Cn(X cochain coefficient cohomology compact surface connected sum continuous map Corollary covering space cup products CW-complex cycles define definition deformation retract denote determine diagram is commutative direct sum disc edges element elementary neighborhood epimorphism equivalence class Euclidean Euler characteristic example Exercises following diagram free abelian group free group free product fundamental group group G hence Hn(X homology class homology theory homomor homomorphism homotopy type Hq(X identity map inclusion map induced homomorphism infinite cyclic group initial point integer isomorphism kernel Klein bottle Lemma Let f Let G map f monomorphism n-cells n-manifold nonorientable normal subgroup notation open set open subsets path class phisms point x0 polygon projective plane Proposition prove quotient group quotient space reader singular n-cube subspace terminal point Theorem 5.1 topological space torus triangles vertices

### References to this book

Computational Homology Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek No preview available - 2004 |