#  ## 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

### 書名 Algebraic topology Hatcher, Allen Cambridge University Press 2001 xii, 544 p. 26 cm 9780521795401 9780521791601 BA55843013 ※クリックでCiNii Booksを表示 英語 イギリス

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