Algebraic TopologyIntended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier. |
Contents
Set theory | 1 |
Modules | 7 |
Categories | 14 |
Homotopy | 22 |
H spaces | 33 |
Suspension | 39 |
The fundamental groupoid | 45 |
Exercises | 56 |
Presheaves | 323 |
Fine presheaves | 329 |
Applications of the cohomology of presheaves | 338 |
Characteristic classes | 346 |
Exact sequences of sets of homotopy classes | 364 |
Higher homotopy groups | 371 |
Change of base points | 379 |
The Hurewicz homomorphism | 387 |
COVERING SPACES AND FIBRATIONS | 64 |
Fiber bundles | 95 |
Exercises | 103 |
POLYHEDRA | 114 |
HOMOLOGY | 156 |
Exercises | 205 |
Homology with coefficients | 212 |
The universalcoefficient theorem for homology | 219 |
The Künneth formula | 227 |
Cohomology | 236 |
Cup and cap products | 248 |
Homology of fiber bundles | 255 |
The cohomology algebra | 263 |
The slant product | 286 |
Duality in topological manifolds | 293 |
The fundamental class of a manifold | 299 |
The Alexander cohomology theory | 306 |
Tautness and continuity | 315 |
The Hurewicz isomorphism theorem | 393 |
CW complexes | 400 |
Homotopy functors | 406 |
Weak homotopy type | 412 |
Exercises | 418 |
EilenbergMacLane spaces | 424 |
Principal fibrations | 432 |
Obstruction theory | 445 |
The suspension map | 452 |
SPECTRAL SEQUENCES AND HOMOTOPY | 464 |
The spectral sequence of a fibration | 473 |
Applications of the homology spectral sequence | 481 |
Multiplicative properties of spectral sequences | 490 |
Applications of the cohomology spectral sequence | 498 |
Serre classes of abelian groups | 504 |
Homotopy groups of spheres | 512 |
Exercises | 518 |
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Common terms and phrases
abelian groups acyclic algebra base point base space bundle chain complex chain equivalence chain map cochain coefficients commutative diagram compact composite continuous map corresponding covariant functor covering projection CW complex defined deformation retract denoted element epimorphism fiber fibration fibration with unique finite follows from theorem fundamental group Given Hausdorff space homology theory homomorphism homotopy classes homotopy groups homotopy type Hq(C inclusion map induces an isomorphism inverse K₁ LEMMA Let Let f locally path connected map f map pair monomorphism morphisms n-manifold neighborhood open covering orientable path component path-connected space pointed space presheaf PROOF Let prove q-simplex relative CW complex short exact sequence simplex simplicial approximation simplicial complex simplicial map simply connected singular singular homology spectral sequence subcomplex subspace THEOREM Let topological space unique path lifting vertex vertices X,xo Y,yo