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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.
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.
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.
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.