Algebraic Topology

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Allen HatcherCambridge University Press, 10/29/2009EAN 9780521795401, ISBN10: 0521795400Paperback, 556 pages, 25.4 x 17.8 x 3.2 cmLanguage: EnglishIn most mathematics departments a t major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology. This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The four main chapters present the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find. Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and a full exposition of Steenrod squares and powers. Researchers will also welcome this aspect of the book.Part I. Some Underlying Geometric Notions1. Homotopy and homotopy type2. Deformation retractions3. Homotopy of maps4. Homotopy equivalent spaces5. Contractible spaces6. Cell complexes definitions and examples7. Subcomplexes8. Some basic constructions9. Two criteria for homotopy equivalence10. The homotopy extension propertyPart II. Fundamental Group and Covering Spaces11. The fundamental group, paths and homotopy12. The fundamental group of the circle13. Induced homomorphisms14. Van Kampen's theorem of free products of groups15. The van Kampen theorem16. Applications to cell complexes17. Covering spaces lifting properties18. The classification of covering spaces19. Deck transformations and group actions20. Additional topicsgraphs and free groups21. K(G,1) spaces22. Graphs of groupsPart III. Homology23. Simplicial and singular homology delta-complexes24. Simplicial homology25. Singular homology26. Homotopy invariance27. Exact sequences and excision28. The equivalence of simplicial and singular homology29. Computations and applications degree30. Cellular homology31. Euler characteristic32. Split exact sequences33. Mayer–Vietoris sequences34. Homology with coefficients35. The formal viewpoint axioms for homology36. Categories and functors37. Additional topics homology and fundamental group38. Classical applications39. Simplicial approximation and the Lefschetz fixed point theoremPart IV. Cohomology40. Cohomology groupsthe universal coefficient theorem41. Cohomology of spaces42. Cup product the cohomology ring43. External cup product44. Poincaré duality orientations45. Cup product46. Cup product and duality47. Other forms of duality48. Additional topics the universal coefficient theorem for homology49. The Kunneth formula50. H-spaces and Hopf algebras51. The cohomology of SO(n)52. Bockstein homomorphisms53. Limits54. More about ext55. Transfer homomorphisms56. Local coefficientsPart V. Homotopy Theory57. Homotopy groups58. The long exact sequence59. Whitehead's theorem60. The Hurewicz theorem61. Eilenberg–MacLane spaces62. Homotopy properties of CW complexes cellular approximation63. Cellular models64. Excision for homotopy groups65. Stable homotopy groups66. Fibrations the homotopy lifting property67. Fiber bundles68. Path fibrations and loopspaces69. Postnikov towers70. Obstruction theory71. Additional topicsbasepoints and homotopy72. The Hopf invariant73. Minimal cell structures74. Cohomology of fiber bundles75. Cohomology theories and omega-spectra76. Spectra and homology theories77. Eckmann-Hilton duality78. Stable splittings of spaces79. The loopspace of a suspension80. Symmetric products and the Dold–Thom theorem81. Steenrod squares and powersAppendixtopology of cell complexesThe compact-open topology.' ... this is a marvellous tome, which is indeed a delight to read. This book is destined to become very popular amongst students and teachers alike.' Bulletin of the Belgian Mathematical Society Show more