Algebraic Topology

by Allen Hatcher

• ISBN13: 9780521795401
• ISBN10: 0521795400
• Format: Paperback
• Copyright: 2001-12-03
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

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. Poincaré

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.