Surfaces

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this free course, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the classification theorem of compact surfaces.

• Introduction
• Learning outcomes
• 1 Topological spaces and homeomorphism
• 1 Topological spaces and homeomorphism
• 2 Examples of surfaces
• 2 Examples of surfaces
• 2.1 Surfaces in space
• 2.2 Surfaces in space
• 2.2.1 Surfaces without boundary
• 2.2.2 Hollow tubing surfaces
• 2.2.3 Surfaces with boundary
• 2.2.4 Boundary numbers
• 2.3 Paper-and-glue constructions
• 2.3.1 Cylinder
• 2.3.2 Möbius band
• 2.3.3 Torus
• 2.3.4 Klein bottle
• 2.3.5 Projective plane
• 2.3.6 Torus with 1 hole
• 2.3.7 Two-fold torus
• 2.3.8 Sphere
• 2.4 Homeomorphic surfaces
• 2.4.1 Remarks
• 2.5 Defining surfaces
• 3 The orientability of surfaces
• 3 The orientability of surfaces
• 3.1 Surfaces with twists
• 3.1.1 Inserting half-twists
• 3.2 Orientability
• 3.2.1 Remarks
• 3.3 The projective plane
• 4 The Euler characteristic
• 4 The Euler characteristic
• 4.1 Nets on surfaces
• 4.2 Subdivisions
• 4.3 The Euler characteristic
• 4.4 Historical note on the Euler characteristic
• 4.5 Some general results
• 4.5.1 Surfaces with holes
• 4.5.2 n-fold toruses
• 4.6 The Classification Theorem
• 4.6.1 Remarks
• 5 Edge identifications
• 5 Edge identifications
• 5.1 Identifying edges of a polygon
• 5.2 The identification topology
• 5.2.1 Proof
• 5.3 Neighbourhoods
• 5.3.1 Torus
• 5.3.2 Klein bottle
• 5.3.3 Torus with 1 hole
• Conclusion
• Acknowledgements