Introduction to Covering Spaces

Topology · Axiom Academy

LESSON Introduction to Covering Spaces Exploring the fundamental concept of covering spaces and their connection to the fundamental group A covering space is one of the most elegant constructions in algebraic topology, providing a bridge between topological spaces and group theory. Let be a topological space. A covering space of consists of a space together with a continuous surjective map such that: For every point , there exists an open neighborhood of such that is a disjoint union of open sets in , each of which is mapped homeomorphically onto by . This definition may seem abstract at first, but it captures a beautiful idea: the covering space "locally looks like" multiple copies of the base space stacked on top of each other. The most fundamental example of a covering space is the real line covering the circle . Think of wrapping the infinite real line around the circle infinitely many times. Each point on the circle has infinitely many preimages in , spaced exactly apart. One of the most powerful features of covering spaces is the path lifting property , which allows us to "lift" paths from the base space to the covering space. Let be a covering map. Given any path in and any point with , there exists a unique path in starting at such that . A path in that winds around once lifts to a path in that goes from some to . If the path winds around times, the lifted path goes from to . The lift is unique once we fix the starting point!

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