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Algebraic Topology

Algebraic Topology


  • Date Published: February 2002
  • availability: Available
  • format: Paperback
  • isbn: 9780521795401

£ 37.99

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About the Authors
  • 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.

    • Broad, readable coverage of the subject
    • Geometric emphasis gives students better intuition
    • Includes many examples and exercises
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    Reviews & endorsements

    '… 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

    '… clear and concise … makes the book useful both as a basis for courses and as a reference work.' Monatshefte für Mathematik

    '… the truly unusual abundance of instructive examples and complementing exercises is absolutely unique of such a kind … the distinctly circumspect, methodologically inductive, intuitive, descriptively elucidating and very detailed style of writing give evidence to the fact that the author's first priorities are exactly what students need when working with such a textbook, namely clarity, readability, steady motivation, guided inspiration, increasing demand, and as much self-containedness of the exposition as possible. No doubt, a very devoted and experienced teacher has been at work here, very much so to the benefit of beginners in the field of algebraic topology, instructors, and interested readers in general.' Zentralblatt MATH

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    Product details

    • Date Published: February 2002
    • format: Paperback
    • isbn: 9780521795401
    • length: 556 pages
    • dimensions: 254 x 178 x 32 mm
    • weight: 0.968kg
    • availability: Available
  • 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.

  • Instructors have used or reviewed this title for the following courses

    • Algebraic Topology
    • Algebraic Topology ll
    • Geometric Topology
    • Introduction to Algebraic Topology
    • Manifolds llI
    • Topology
  • Author

    Allen Hatcher, Cornell University, New York

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