Unwrapping spaces, path lifting, and the Galois correspondence
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.
Watch a covering space peel away from its base. Use the peel sliderto separate the cover from the base circle, and the deck transformationbutton to apply the covering automorphism — shifting the helix by one level or rotating the n-fold cover by 2π/n.
The real line wraps helically over the circle
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)).
Every path in the base space lifts uniquely to the covering space once you fix a starting point. Click on the base circle to draw a path and watch it lift to the helix in real time. Try the presets to see how contractible loopslift to loops, while generators of π&sub1; lift to paths that end on a different sheet.
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.
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. Click nodes in the lattice to explore which subgroup of ℤ corresponds to which cover of S¹.
Click a node to see the corresponding covering space. Larger subgroups correspond to 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.