Lectures on Algebraic TopologySpringer Science & Business Media, 6 déc. 2012 - 380 pages This is essentially a book on singular homology and cohomology with special emphasis on products and manifolds. It does not treat homotopy theory except for some basic notions, some examples, and some applica tions of (co-)homology to homotopy. Nor does it deal with general(-ised) homology, but many formulations and arguments on singular homology are so chosen that they also apply to general homology. Because of these absences I have also omitted spectral sequences, their main applications in topology being to homotopy and general (co-)homology theory. Cech cohomology is treated in a simple ad hoc fashion for locally compact subsets of manifolds; a short systematic treatment for arbitrary spaces, emphasizing the universal property of the Cech-procedure, is contained in an appendix. The book grew out of a one-year's course on algebraic topology, and it can serve as a text for such a course. For a shorter basic course, say of half a year, one might use chapters II, III, IV (§§ 1-4), V (§§ 1-5, 7, 8), VI (§§ 3, 7, 9, 11, 12). As prerequisites the student should know the elementary parts of general topology, abelian group theory, and the language of categories - although our chapter I provides a little help with the latter two. For pedagogical reasons, I have treated integral homology only up to chapter VI; if a reader or teacher prefers to have general coefficients from the beginning he needs to make only minor adaptions. |
Table des matières
1 | |
Chapter II | 16 |
Chapter III | 29 |
Invariance under Homotopy | 37 |
Small Simplices Excision | 43 |
7 Jordan Theorem Invariance of Domain | 78 |
Euclidean Neighborhood Retracts ENRs | 79 |
Cellular Decomposition and Cellular Homology | 85 |
The Interior Homology Product Pontrjagin Product | 193 |
Intersection Numbers in IR | 197 |
The Fixed Point Index | 202 |
The LefschetzHopf Fixed Point Theorem | 207 |
The Exterior Cohomology Product | 214 |
The Interior Cohomology Product Product | 219 |
9Products in Projective Spaces Hopf Maps and Hopf Invariant | 222 |
Hopf Algebras | 227 |
2 CWSpaces | 88 |
Examples | 95 |
Homology Properties of CWSpaces | 101 |
5 The EulerPoincaré Characteristic | 104 |
Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism | 106 |
Simplicial Spaces | 111 |
Simplicial Homology | 119 |
Functors of Complexes | 123 |
Additive Functors | 127 |
3 Derived Functors | 132 |
Universal Coefficient Formula | 136 |
Tensor and Torsion Products | 140 |
Hom and Ext | 146 |
Singular Homology and Cohomology with General Coefficient Groups | 150 |
Tensorproduct and Bilinearity | 157 |
Tensorproduct of Complexes Künneth Formula | 161 |
Hom of Complexes Homotopy Classification of Chain Maps | 167 |
Acyclic Models | 174 |
The EilenbergZilber Theorem Künneth Formulas for Spaces | 178 |
Products | 186 |
The Scalar Product | 187 |
The Exterior Homology Product | 189 |
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 |
Homology of Dimensions n in nManifolds | 259 |
Fundamental Class and Degree | 266 |
Limits | 272 |
Čech Cohomology of Locally Compact Subsets of IR | 283 |
PoincaréLefschetz Duality | 291 |
Examples Applications | 298 |
Duality in aManifolds | 303 |
Transfer | 308 |
Thom Class Thom Isomorphism | 314 |
The Gysin Sequence Examples | 325 |
Intersection of Homology Classes | 335 |
Appendix Kan and ČechExtensions of Functors | 348 |
Polyhedrons under a Space and Partitions of Unity | 352 |
3 Extending Functors from Polyhedrons to More General Spaces | 361 |
368 | |
371 | |
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Expressions et termes fréquents
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