Universal Covering Spaces in Algebraic Topology

Explore the concept of universal covering spaces in algebraic topology through this comprehensive lecture. Examine examples of the main theorem from the previous lecture, demonstrating how the associated homomorphism of fundamental groups injects pi(X) as a subgroup of pi(B) in covering spaces. Investigate helical coverings of a circle and a two-fold covering of the wedge of two circles. Learn how covering spaces of a space B are associated with subgroups of pi(B), and discover the properties of the universal covering space. Follow the construction of a universal cover using paths from a fixed base point, and explore its applications to various geometric surfaces. Gain insights into the connection between geometry and topology, and understand how other covering spaces can be derived from a universal covering space.