Sumarios

• Part I - Fundamental group and covering spaces
  • 1. Categories and Functors.
  • 2. The fundamental group.
  • 3. $\pi_1(S^1)$ and applications.
  • 4. Generators and relations; Siefert-van Kampen's theorem.
  • 5. Realization theorem; Homotopy of cell complexes.
  • 6. Liftings of maps and covering spaces
  • 7. Coverings and subgroups of $\pi_1$
  • 8. Homotopy type and higher homotopy groups.

• Part II - Homology
  • 9. Delta homology and Singular Homology.
  • 10. Homotopy invariance of Homology.
  • 11. $H_1$ and $\pi_1$
  • 12. Exact sequence of a pair
  • 13. The excision theorem.
  • 14. Equivalence between Homologies.
  • 15. Invariance of dimension.
  • 16. Mayer-Vietoris theorem.
  • 17. Degree and Euler characteristic.

• Part III - Cohomology and Duality
  • 18. Universal coefficient theorem.
  • 19. Long exact sequence of a pair.
  • 20. Cup product formula.
  • 21. Kunneth formula.
  • 22. $H_n$, orientation and compactness.
  • 23. Poincaré duality.
  • 24. (Co)homology with compact support.

• Part IV - Higher Homotopy Theory
  • 25. Exact sequences in homotopy.
  • 26. Whitehead's theorem.
  • 27. Excision for homotopy; homotopy type for K(G,n) spaces.
  • 28. Hurewicz's theorem.