Contents:
Introduction to Algebraic Topology – One-dimensional objects – Homeomorphism and the group structure on a circle – Two-dimensional surfaces: the sphere – More on the sphere – Two-dimensional objects – the torus and genus – Non-orientable surfaces – the Mobius band – The Klein bottle and projective plane – Polyhedra and Euler’s formula

Applications of Euler’s formula and graphs – More on graphs and Euler’s formula – Rational curvature, winding and turning – Duality for polygons and the Fundamental Theorem of Algebra – More applications of winding numbers – rational curvature of a polytope – Rational curvature of polytopes and the Euler number – The geometry of surfaces – The two-holed torus and 3-crosscaps surface – Knots and surfaces I

### Course Curriculum

 AlgTop0: Introduction to Algebraic Topology Details 30:6 AlgTop1: One-dimensional objects Details 32:19 AlgTop2: Homeomorphism and the group structure on a circle Details 52:9 AlgTop3: Two-dimensional surfaces: the sphere Details 42:27 AlgTop4: More on the sphere Details 40:51 AlgTop5: Two-dimensional objects- the torus and genus Details 49:33 AlgTop6: Non-orientable surfaces—the Mobius band Details 42:5 AlgTop7: The Klein bottle and projective plane Details 39:42 AlgTop8: Polyhedra and Euler’s formula Details 45:35 AlgTop9: Applications of Euler’s formula and graphs Details 42:23 AlgTop10: More on graphs and Euler’s formula Details 47:50 AlgTop11: Rational curvature, winding and turning Details 48:19 AlgTop12: Duality for polygons and the Fundamental Theorem of Algebra Details 45:36 AlgTop13: More applications of winding numbers Details 26:59 AlgTop15: Rational curvature of a polytope Details 50:23 AlgTop16: Rational curvature of polytopes and the Euler number Details 35:28 AlgTop20: The geometry of surfaces Details 43:54 AlgTop21: The two-holed torus and 3-crosscaps surface Details 38:53 AlgTop22: Knots and surfaces I Details 52:23

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