Delve into advanced concepts of spherical geometry and complex analysis in this fourth lecture of the Algebraic Topology series. Explore inversive geometry on both the plane and sphere, and discover the Riemann sphere model of the complex plane with a point at infinity. Learn about the sphere as the projective line over rational complex numbers, and gain insights into Riemann surfaces and their topological explanations. Examine transformations of the Riemann sphere and understand the connection between inversive geometry and complex analysis. Suitable for those with a strong foundation in mathematics seeking to expand their knowledge of algebraic topology and complex geometry.