Lectures on Algebraic TopologySpringer is reissuing a selected few highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. Springer-Verlag began publishing books in higher mathematics in 1920. This is a reprint of the Second Edition. |
Contents
Preliminaries on Categories Abelian Groups and Homotopy | 1 |
Abelian Groups Exactness Direct Sums Free Abelian Groups | 7 |
Homotopy | 13 |
Homology of Complexes | 16 |
Connecting Homomorphism Exact Homology Sequence | 19 |
ChainHomotopy | 23 |
Free Complexes | 26 |
Singular Homology | 29 |
Singular Homology and Cohomology with General Coefficient Groups | 150 |
Tensorproduct and Bilinearity | 157 |
Tensorproduct of Complexes Kiinneth Formula | 161 |
Horn of Complexes Homotopy Classification of Chain Maps | 167 |
11 Acyclic Models | 174 |
Kiinneth Formulas for Spaces | 178 |
Products | 186 |
The Scalar Product | 187 |
The Singular Complex | 30 |
Singular Homology | 32 |
Special Cases | 33 |
Invariance under Homotopy | 37 |
Barycentric Subdivision | 40 |
Small Simplices Excision | 43 |
MayerVietoris Sequences | 47 |
Applications to Euclidean Space | 54 |
Homology of Cells and Spheres | 55 |
Local Homology | 59 |
The Degree of a Map | 62 |
Local Degrees | 66 |
Homology Properties of Neighborhood Retracts in IR | 71 |
Jordan Theorem Invariance of Domain | 78 |
Euclidean Neighborhood Retracts ENRs | 79 |
Cellular Decomposition and Cellular Homology | 85 |
2 CWSpaces | 88 |
3 Examples | 95 |
Homology Properties of CWSpaces | 101 |
The EulerPoincare Characteristic | 104 |
Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism | 106 |
Simplicial Spaces | 111 |
8 Simplicial Homology | 119 |
Functors of Complexes | 123 |
2 Additive Functors | 127 |
Derived Functors | 132 |
4 Universal Coefficient Formula | 136 |
Tensor and Torsion Products | 140 |
Horn and Ext | 146 |
2 The Exterior Homology Product | 189 |
The Interior Homology Product Pontrjagin Product | 193 |
Intersection Numbers in R | 197 |
The Fixed Point Index | 202 |
The LefschetzHopf Fixed Point Theorem | 207 |
The Exterior Cohomology Product | 214 |
The Interior Cohomology Product vProduct | 219 |
Products in Projective Spaces Hopf Maps and Hopf Invariant | 222 |
10 Hopf Algebras | 227 |
11 The Cohomology Slant Product | 233 |
The CapProduct Product | 238 |
The Homology Slant Product and the Pontrjagin Slant Product | 245 |
Manifolds | 247 |
The Orientation Bundle of a Manifold | 251 |
Fundamental Class and Degree | 266 |
5 Limits | 272 |
Cech Cohomology of Locally Compact Subsets of JR | 281 |
Poincar6Lefschetz Duality | 291 |
Examples Applications | 298 |
9 Duality in 9Manif olds | 303 |
10 Transfer | 308 |
11 Thom Class Thom Isomorphism | 315 |
The Gysin Sequence Examples | 325 |
Intersection of Homology Classes | 335 |
Appendix Kan and CechExtensions of Functors | 348 |
Polyhedrons under a Space and Partitions of Unity | 352 |
to More General Spaces | 361 |
Bibliography | 368 |
| 371 | |
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Common terms and phrases
abelian group additive agrees algebra apply arbitrary assertion assume base boundary called cells chain map characteristic choose Clearly closed coefficients cohomology commutative compact complex composite connected consider consists contained continuous Corollary covering define Definition deformation denote depends diagram dimension direct sum elements exact sequence example excisive Exerc Exercises exists extension fact finite fixed follows formula functions functor given gives graded hence holds homology homology class homomorphism identifying implies inclusion induces instance intersection isomorphic Lemma limit linear locally manifold modules morphism multiplication natural neighborhood neighborhood retract obtained open set oriented pairs particular projective Proof properties Proposition prove R-module Remark representative resp respect result ring Similarly simplicial space subset takes theorem topological transformation unique universal vertices write zero