Hjem

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.