Contents vii

Lecture 12 80

a. Non-orientable surfaces and M¨ obius caps 80

b. Calculation of Euler characteristic 81

c. Covering non-orientable surfaces 83

d. Classification of orientable surfaces 85

Lecture 13 86

a. Proof of the classification theorem 86

b. Non-orientable surfaces: Classification and models 91

Lecture 14 92

a. Chain complexes and Betti numbers 92

b. Homology of surfaces 94

c. A second interpretation of Euler characteristic 96

Lecture 15 98

a. Interpretation of the Betti numbers 98

b. Torsion in the first homology and non-orientability 100

c. Another derivation of interpretation of Betti numbers 101

Chapter 3. Differentiable Structure on Surfaces: Real and

Complex 103

Lecture 16 103

a. Charts and atlases 103

b. First examples of atlases 106

Lecture 17 109

a. Differentiable manifolds 109

b. Diffeomorphisms 110

c. More examples of charts and atlases 113

Lecture 18 117

a. Embedded surfaces 117

b. Gluing surfaces 117

c. Quotient spaces 118

d. Removing singularities 120

Lecture 19 121

a. Riemann surfaces: Definition and first examples 121

b. Holomorphic equivalence of Riemann surfaces 125