Algebraic topology

Allen Hatcher

In most mathematics departments at 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.

「Nielsen BookData」より

[目次]

  • Part I. Some Underlying Geometric Notions: 1. Homotopy and homotopy type
  • 2. Deformation retractions
  • 3. Homotopy of maps
  • 4. Homotopy equivalent spaces
  • 5. Contractible spaces
  • 6. Cell complexes definitions and examples
  • 7. Subcomplexes
  • 8. Some basic constructions
  • 9. Two criteria for homotopy equivalence
  • 10. The homotopy extension property
  • Part II. Fundamental Group and Covering Spaces: 11. The fundamental group, paths and homotopy
  • 12. The fundamental group of the circle
  • 13. Induced homomorphisms
  • 14. Van Kampen's theorem of free products of groups
  • 15. The van Kampen theorem
  • 16. Applications to cell complexes
  • 17. Covering spaces lifting properties
  • 18. The classification of covering spaces
  • 19. Deck transformations and group actions
  • 20. Additional topics: graphs and free groups
  • 21. K(G,1) spaces
  • 22. Graphs of groups
  • Part III. Homology: 23. Simplicial and singular homology delta-complexes
  • 24. Simplicial homology
  • 25. Singular homology
  • 26. Homotopy invariance
  • 27. Exact sequences and excision
  • 28. The equivalence of simplicial and singular homology
  • 29. Computations and applications degree
  • 30. Cellular homology
  • 31. Euler characteristic
  • 32. Split exact sequences
  • 33. Mayer-Vietoris sequences
  • 34. Homology with coefficients
  • 35. The formal viewpoint axioms for homology
  • 36. Categories and functors
  • 37. Additional topics homology and fundamental group
  • 38. Classical applications
  • 39. Simplicial approximation and the Lefschetz fixed point theorem
  • Part IV. Cohomology: 40. Cohomology groups: the universal coefficient theorem
  • 41. Cohomology of spaces
  • 42. Cup product the cohomology ring
  • 43. External cup product
  • 44. Poincare duality orientations
  • 45. Cup product
  • 46. Cup product and duality
  • 47. Other forms of duality
  • 48. Additional topics the universal coefficient theorem for homology
  • 49. The Kunneth formula
  • 50. H-spaces and Hopf algebras
  • 51. The cohomology of SO(n)
  • 52. Bockstein homomorphisms
  • 53. Limits
  • 54. More about ext
  • 55. Transfer homomorphisms
  • 56. Local coefficients
  • Part V. Homotopy Theory: 57. Homotopy groups
  • 58. The long exact sequence
  • 59. Whitehead's theorem
  • 60. The Hurewicz theorem
  • 61. Eilenberg-MacLane spaces
  • 62. Homotopy properties of CW complexes cellular approximation
  • 63. Cellular models
  • 64. Excision for homotopy groups
  • 65. Stable homotopy groups
  • 66. Fibrations the homotopy lifting property
  • 67. Fiber bundles
  • 68. Path fibrations and loopspaces
  • 69. Postnikov towers
  • 70. Obstruction theory
  • 71. Additional topics: basepoints and homotopy
  • 72. The Hopf invariant
  • 73. Minimal cell structures
  • 74. Cohomology of fiber bundles
  • 75. Cohomology theories and omega-spectra
  • 76. Spectra and homology theories
  • 77. Eckmann-Hilton duality
  • 78. Stable splittings of spaces
  • 79. The loopspace of a suspension
  • 80. Symmetric products and the Dold-Thom theorem
  • 81. Steenrod squares and powers
  • Appendix: topology of cell complexes
  • The compact-open topology.

「Nielsen BookData」より

この本の情報

書名 Algebraic topology
著作者等 Hatcher, Allen
出版元 Cambridge University Press
刊行年月 2001
ページ数 xii, 544 p.
大きさ 26 cm
ISBN 9780521795401
9780521791601
NCID BA55843013
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言語 英語
出版国 イギリス
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