Hatcher-Algebraic-Topology · Problem 1.3.4
The Fundamental Group
Mathematics — Graph Theory · covering-spaces
Problem
Construct a simply-connected covering space of the space that is the union of a sphere and a diameter. Do the same when is the union of a sphere and a circle intersecting it in two points.Intuitive explanation
The key insight is that a covering space "unwinds" the loops in the original space. For the sphere-plus-diameter, the diameter creates a loop that can't be contracted—imagine walking along the diameter and back; you need to track which "copy" of the sphere you're on. The universal cover duplicates and arranges these copies so every loop becomes contractible. For the sphere-plus-circle case, the intersecting circle adds a second independent loop, so you'll need to unwind both simultaneously by creating a grid-like structure of copies. Think of it as building a simply-connected "staircase" version of where you can always retract paths to a point without getting trapped by the original topology.
Intuitive explanation
The key insight is that a covering space "unwinds" the loops in the original space. For the sphere-plus-diameter, the diameter creates a loop that can't be contracted—imagine walking along the diameter and back; you need to track which "copy" of the sphere you're on. The universal cover duplicates and arranges these copies so every loop becomes contractible. For the sphere-plus-circle case, the intersecting circle adds a second independent loop, so you'll need to unwind both simultaneously by creating a grid-like structure of copies. Think of it as building a simply-connected "staircase" version of where you can always retract paths to a point without getting trapped by the original topology.