## 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 |

6 Triangulations of Compact Surfaces | 14 |

8 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 |

7 The Fundamental Group of a Product Space | 52 |

8 Homotopy Type and Homotopy Equivalence of Spaces | 54 |

CHAPTER VIII | 186 |

3 Homology of Finite Graphs | 192 |

4 Homology of Compact Surfaces | 201 |

5 The MayerVietoris Exact Sequence | 207 |

6 The JordanBrouwer Separation Theorem and Invariance of Domain | 211 |

7 The Relation between the Fundamental Group and the First Homology Group | 217 |

References | 224 |

Homology of CWCompIexes | 225 |

References | 59 |

CHAPTER HI Free Groups and Free Products of Groups | 60 |

3 Free Abelian Groups | 63 |

4 Free Products of Groups | 71 |

5 Free Groups | 75 |

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

7 Universal Mapping Problems | 81 |

References | 85 |

Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces Applications | 86 |

2 Statement and Proof of the Theorem of Seifert and Van Kampen | 87 |

3 First Application of Theorem 2 1 | 91 |

4 Second Application of Theorem 2 1 | 95 |

5 Structure of the Fundamental Group of a Compact Surface | 96 |

6 Application to Knot Theory | 103 |

7 Proof of Lemma 2 4 | 108 |

References | 116 |

Covering Spaces | 117 |

3 Lifting of Paths to a Covering Space | 123 |

4 The Fundamental Group of a Covering Space | 126 |

5 Lifting of Arbitrary Maps to a Covering Space | 127 |

6 Homomorphisms and Automorphisms of Covering Spaces | 130 |

7 The Action of the Group nXx on the Set pl x | 133 |

8 Regular Covering Spaces and Quotient Spaces | 135 |

The BorsukUlam Theorem for the 2Sphere | 138 |

10 The Existence Theorem for Covering Spaces | 140 |

References | 146 |

CHAPTER VI | 147 |

3 Some Examples of Problems which Motivated the Development of Homology Theory in the Nineteenth Century | 149 |

References | 157 |

CHAPTER VII | 158 |

3 The Homomorphism Induced by a Continuous Map | 163 |

4 The Homotopy Property of the Induced Homomorphisms | 166 |

5 The Exact Homology Sequence of a Pair | 169 |

6 The Main Properties of Relative Homology Groups | 173 |

7 The Subdivision of Singular Cubes and the Proof of Theorem 6 4 | 178 |

3 CWComplexes | 228 |

4 The Homology Groups of a CWComplex | 232 |

5 Incidence Numbers and Orientations of Cells | 238 |

6 Regular CWComplexes | 243 |

7 Determination of Incidence Numbers for a Regular Cell Complex | 244 |

8 Homology Groups of a Pseudomanifold | 249 |

References | 253 |

CHAPTER x | 254 |

3 Definition and Basic Properties of Homology with Arbitrary Coefficients | 262 |

4 Intuitive Geometric Picture of a Cycle with Coefficients in G | 266 |

5 Coefficient Homomorphisms and Coefficient Exact Sequences | 267 |

6 The Universal Coefficient Theorem | 269 |

7 Further Properties of Homology with Arbitrary Coefficients | 274 |

References | 278 |

CHAPTER XI | 279 |

2 The Product of CWComplexes and the Tensor Product of Chain Complexes | 280 |

3 The Singular Chain Complex of a Product Space | 282 |

4 The Homology of the Tensor Product of Chain Complexes The Kiinneth Theorem | 284 |

5 Proof of the EilenbergZilber Theorem | 286 |

6 Formulas for the Homology Groups of Product Spaces | 300 |

References | 303 |

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 |

CHAPTER XV | 394 |

4 The Hopf Invariant | 402 |

2 Differentiable Singular Chains | 408 |

APPENDIX | 419 |

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### Common terms and phrases

algebraic algebraic topology apply assert Assume boundary called chain complexes chain homotopy chain map Chapter choose circle clear closed coefficient cohomology compact complex components condition connected connected sum consider consisting contained continuous map Corollary corresponding covering space CW-complex cycles define definition denote described determine diagram direct disc edges element equivalence exact sequence example Exercises exists fact Figure finite following diagram formula free abelian group free group function fundamental group give given hence hold homology groups homomorphism homotopy identified identity inclusion map integer Introduction isomorphism Lemma manifold n-dimensional natural neighborhood Note obtain obvious operator orientable pair path problem projective plane proof properties Proposition prove reader relation represented respectively result retract sequence simple singular statement structure subgroup subset surface theorem theory topological topological space torus triangles unique vertices

### References to this book

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