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The Algebraic Topology: A Beginner's Course

The Algebraic Topology: A Beginner's Course (UNSW). Taught by N. J. Wildberger, this course provides an introduction to algebraic topology, with emphasis on visualization, geometric intuition and simplified computations. Algebraic topology is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This course introduces a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. And it also has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical.

We continue computing homology groups. Here we tackle the 2-sphere. First we compute the homology using the model of a tetrahedron: four 2-dimensional faces, but no 3-dim solid. This illustrates how linear algebra naturally arises in this kind of problem. We then provide a much simpler alternative calculation using the more flexible framework of semi-simplicial complexes, or delta-complexes.

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