Algebraic Topology

Produktbeschreibung

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. 

Inhaltsverzeichnis

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. 

Kritik

'... 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 

Autoreninfo

Cornell University, New York