Algebraic Topology

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: 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 is the inclusion of many optional topics not usually part of a first course due to time constraints: 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 Steenrod squares and powers.

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.

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.

This introductory textbook is suitable for use in a first-year graduate course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. Along with the basic material on fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory, the book includes many optional topics for which elementary expositions are hard to find.