# Topology

## Syllabus

Revised May 2006.

The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.

### General Topology

• Topological spaces, continuous functions, product and quotient topology [1, ch. 2]
• Connectedness and compactness [1, ch. 3]
• Countability and separation axioms, Urysohn lemma, Tietze theorem [1, ch. 4, except §36]
• Complete metric spaces and function spaces [1, §43, 45]

### Algebraic Topology

• Classification of surfaces [2, ch. I]
• Fundamental group [2, ch. II], [3, §1.1]
• van Kampen’s theorem [2, ch. III, IV], [3, §1.2]
• Classification of covering spaces [2, ch. V], [3 §1.3]

### Homology:

• Simplicial, singular, cellular; computations and applications [3, ch. 2], [4, ch. 4]
• Degree of a map $$S^n\to S^n$$ [3, p. 134], [4, §21]
• Euler characteristic [3, p. 146]
• Lefschetz fixed point theorem [3, p. 179], [4, §22]

# References

• [1] J. Munkres, Topology , second edition, Prentice-Hall, 2000.
• [2] W. Massey, A Basic Course in Algebraic Topology , Springer-Verlag, 1991.
• [3] A. Hatcher, Algebraic Topology , Cambridge U. Press, 2002.
• [4] J. Munkres, Elements of Algebraic Topology , Addison-Wesley, 1984.