Classification of Surfaces: Bachelor Thesis

Classification of Surfaces
Bachelor Thesis
Ana da Silva Rodrigues
June, 2023
Advisors: Prof. Dr. Ana Cannas da Silva
Department of Mathematics, ETH Z¨urich

Aknowledgments
I thank my supervisor Ana Cannas da Silva for her guidance
and support throughout the whole process. She chose a topic
matching my interests and stood by me the whole time giving me
enough space to follow my way, while kindly guiding me when
needed. I appreciate her effort to take time to satiate my curiosity,
even when my questions diverged from the main topic. The way
she sees mathematics inspires me and I hope that the style of this
thesis reflects a bit of the joy she transmits.
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Contents
Contents iii
1 Compact surfaces 5
1.1 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Orientability............................. 7
2 Triangulation and polygonal presentation 9
2.1 Triangulation of compact surfaces . . . . . . . . . . . . . . . . 9
2.2 Labelling scheme of polygons . . . . . . . . . . . . . . . . . . . 12
2.3 Representation of the standard surfaces . . . . . . . . . . . . . 13
2.3.1 Thesphere.......................... 13
2.3.2 Thetorus........................... 13
2.3.3 The projective plane . . . . . . . . . . . . . . . . . . . . 14
2.3.4 Triangulation of the standard surfaces . . . . . . . . . . 15
2.4 The Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . 15
3 Connected sums 19
3.1 Euler characteristic of connected sums . . . . . . . . . . . . . . 20
3.2 Connected sum of spheres . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 A sphere connected to an arbitrary surface . . . . . . . 20
3.3 Connected sum of tori . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Connected sum of projective planes . . . . . . . . . . . . . . . 22
3.5 The M¨ obius strip in the projective plane . . . . . . . . . . . . . 23
3.6 The Klein bottle or two connected projective planes . . . . . . 23
3.7 A torus connected to a projective plane . . . . . . . . . . . . . 24
3.8 The monoid of compact surfaces up to homeomorphism . . . 27
4 Classification of surfaces 31
4.1 Part I: Classification of a surface . . . . . . . . . . . . . . . . . 32
4.1.1 Step 1: Polygonal presentation of the surface . . . . . . 32
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