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

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