Covering Space - Examples

Examples

Consider the unit circle S1 ⊂ R2. The map p : R → S1 with

is a cover where each point of S1 is covered infinitely often.

As another example, take the complex plane with the origin removed (denoted by C*, and pick a non-zero integer n. Then q_n : C* → C* given by

is a cover. Here, every fiber has n elements. The map q_n leaves the unit circle S1 invariant, and the restriction of this map to S1 is an n-fold cover of the circle by itself.

In fact, S1 and R are the only connected covering spaces of the circle. To prove this, we first note that the fundamental group of the circle is isomorphic to the additive group of integers Z. As follows from the correspondence between equivalence classes of connected coverings and conjugacy classes of subgroups of the fundamental group of the base space discussed below, a connected covering f : C → S1 is determined by a subgroup f_#(π₁(C)) of π₁(S1) = Z, where f_# is the induced homomorphism. The group Z is abelian and it only has two kinds of subgroups: the trivial subgroup (which has infinite subgroup index in Z) and the subgroups H_n = {kn : k ∈ Z} for n = 1, 2, 3, ..., where H_n has index n in Z. Each of the subgroups H_n of Z is realized by the covering q_n : S1 → S1 since one can check that (q_n)_# : Z→ Z maps an integer k to kn and hence, (q_n)_#(Z) = H_n. The trivial subgroup of Z is realized by the covering p : R → S1, since R is simply connected and has trivial fundamental group and hence p_#(π₁(R)) = {0}, the trivial subgroup of Z. Since the total space of the coverings q_n is S1 and the total space of the covering p is R, this shows that every connected cover of S1 is either S1 or R.

A further example, originating from physics (see quantum mechanics), is the special orthogonal group SO(3) of rotations of R3, which has the "double" covering group SU(2) of unitary rotations of C2 (in quantum mechanics acting as the group of spinor rotations). Both groups have identical Lie algebras, but only SU(2) is simply connected.