Syllabus:
Part 1 - Fundamental Group and Covering Spaces
1. Categories and Functors, Topological spaces and natural operations on them.
2. The fundamental group. First examples and consequences.
3. Van Kampen's Theorem, cell complexes and applications.
4. Lifting properties, Covering spaces, K(G,1) spaces.
Part 2 - Homology
5. Complexes and their homology. Singular homology and its properties. Homotopy Invariance.
6. Exact Sequences and Excision. Equivalence between homologies.
7. First Homology and Fundamental Group. Classical applications.
8. Mayer-Vietoris Sequences, Degree, Euler characteristic.
Part 3 - Cohomology
9. The Universal Coefficient Theorem. Cohomology of Spaces
10. Cohomology Ring. Kunneth Formula. Spaces with Polynomial Cohomology.
11. Orientations and Homology. The Duality Theorem. Cup Product and Duality.
12. Universal Coefficient Theorem for Homology. The General Kunneth Formula. The Cohomology of SO(n)
Part 4 - Higher Homotopy Theory
13. Whitehead's Theorem. CW Approximation.
14. Excision for Homotopy. The Hurewicz Theorem. Fiber Bundles.
Recommended Bibliography:
Allen Hatcher, "Algebraic Topology", available in https://www.math.cornell.edu/~hatcher/AT/ATpage.html .