Unwrapping spaces, path lifting, and the Galois correspondence
Unwind a circle and you get an endless helix. Covering spaces unwrap a space layer by layer, and the way loops lift into the cover reveals the space's fundamental group.
Pick a cover and drag the peel slider to lift it off the base circle. Press deck transformation to shift the helix up one level, or turn an n-fold cover by 1/n of a circle.
The real line wraps helically over the circle
A covering space is a space that maps onto another space by “evenly covering” it — locally the map looks like stacking identical copies, but globally the covering space may have a very different shape. The classic example: the real line ℝ wraps helically over the circle S¹, unwinding it into something simply connected.
Covering spaces are intimately connected to the fundamental group. Paths in the base space lift uniquely to the cover, and whether a loop lifts to a loop or an open path tells you everything about π&sub1;. The crowning result is the Galois correspondence: there is a bijection between covering spaces and subgroups of the fundamental group, paralleling the classical correspondence in field theory.
Key insight: The deck transformations of a covering form a group that acts freely on each fiber. For the universal cover ℝ → S¹, the deck group is ℤ (integer shifts of the helix). For the n-fold cover, it is ℤ/nℤ (rotations by multiples of 2π/n). This group is isomorphic to π&sub1;(base) / pₚ(π&sub1;(cover)).
Click points on the base circle to draw a path and watch it lift to the helix, or load a preset. A loop that doesn't wind around comes back to its start upstairs; a loop that goes once around ends on the next level up.
Every path in the base space lifts uniquely to the covering space once you fix a starting point.
Key insight: A loop in the base lifts to a loop in the cover if and only if it represents an element of the subgroup pₚ(π&sub1;(cover)) ≤ π&sub1;(base). For the universal cover, only contractible loops lift to loops — because π&sub1;(ℝ) is trivial. This is how covering spaces “detect” the fundamental group.
Click a subgroup of ℤ in the lattice to see the covering of the circle that matches it. Notice the order flips: bigger subgroups give smaller covers.
Click a node to see the corresponding covering space. Larger subgroups correspond to smaller (more folded) covers.
The classification theorem for covering spaces establishes a lattice-reversing bijection: subgroups of π&sub1; correspond to covering spaces, with bigger subgroups giving smaller (more folded) covers.
Key insight: The trivial subgroup {0} corresponds to the universal cover ℝ, which is the “largest” covering space. The whole group ℤ corresponds to the identity cover. Subgroups nℤ of index n give exactly the n-fold covers. This mirrors the Galois correspondence between subgroups and intermediate field extensions.
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