The Spiral Over the Circle

Algebraic Topology · Axiom Academy

INTRO The Spiral Over the Circle Explore the canonical covering space: the real line wrapping around the circle. Adjust the slider to see how real numbers on the line wrap around the circle via the map . Visualize the entire real line as it wraps infinitely many times around the circle. Select a point on the circle to see all the real numbers that map to it - the fiber over that point. Zoom into a neighborhood to see how the covering map is locally a homeomorphism. Explore the symmetries of the covering space that preserve the covering map. The covering is universal because is simply connected. Every other connected covering space of can be obtained as a quotient of this one. The deck transformations form a group isomorphic to . The covering space encodes the fundamental group geometrically through its symmetries. 1. Surjective: Every point on the circle is covered. 2. Local homeomorphism: Small neighborhoods map bijectively. 3. Evenly covered: Each neighborhood lifts to disjoint copies. 4. Universal: The covering space is simply connected. This canonical example generalizes to compact Lie groups (their universal covers are simply connected Lie groups), Riemann surfaces (their universal covers are the disk, plane, or sphere), and many other contexts in topology and geometry!

This is the written version of the interactive lesson above. See the full Algebraic Topology course.