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. |
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 |
5 The MayerVietoris Exact Sequence | 207 |
7 The Relation between the Fundamental Group | 217 |
Homology of CWComplexes | 225 |
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 |
References | 59 |
CHAPTER III | 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 |
8 Regular Covering Spaces and Quotient Spaces | 135 |
CHAPTER VI | 147 |
CHAPTER VII | 158 |
4 The Homotopy Property of the Induced Homomorphisms | 166 |
6 The Main Properties of Relative Homology Groups | 173 |
CHAPTER VIII | 186 |
3 Homology of Finite Graphs | 192 |
4 Homology of Compact Surfaces | 201 |
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 Künneth 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 Poincaré 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 |
Other editions - View all
Common terms and phrases
A₁ algebraic topology arcwise connected Assume chain complexes chain homotopy chain map Chapter circle closed path cochain coefficient cohomology commutative diagram compact surface connected sum consider continuous map Corollary covering space cup products CW-complex cycles d₁ define definition deformation retract denote diagram is commutative disc E₁ edges element equivalence class Euclidean example EXERCISES exists f₁ finite following diagram free abelian group free group free product fundamental group G₁ group G H₁(X H₄(X hence Ho(X homology class homology groups homology sequence homology theory homomorphism identity map inclusion map induced homomorphism initial point integer isomorphism kernel Lemma Let f manifold map f monomorphism n-dimensional nonorientable obtain open subsets orientable p₁ pair path class phisms Poincaré projective plane proof Proposition prove quotient group quotient space reader singular n-cube terminal point Theorem 5.1 topological space torus triangles U₁ unique vertices x₁ αι
References to this book
Computational Homology Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek No preview available - 2004 |