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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 illustrate the idea of a covering space by looking at the rich examples coming from a wedge of two circles. Coverings of this space are graphs with each vertex of degree four, with edges suitably labelled in a directed way with alpha's and beta's. We also introduce the idea of a universal covering space, which is by definition simply connected, or equivalently its fundamental group is trivial, and illustrate in the case of the wedge of circles.

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