Covering group

Covering group

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : GH is a continuous group homomorphism. The map p is called the covering homomorphism.

Properties

Let G be a covering group of H. The kernel K of the covering homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff (and if and only if H is Hausdorff). Going in the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p : GG/K is a covering homomorphism.

If G is connected then K, being a discrete normal subgroup, necessarily lies in the center of G and is therefore abelian. In this case, the center of H = G/K is given by

Z(H) cong Z(G)/K.

As with all covering spaces, the fundamental group of G injects into the fundamental group of H. If G is path-connected then the quotient group pi_1(H)/pi_1(G) is isomorphic to K. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. The group K acts simply transitively on the fibers (which are just left cosets) by right multiplication. The group G is then a principal K-bundle over H.

If G is a covering group of H then the groups G and H are locally isomorphic. Moreover, given any two connected locally isomorphic groups H1 and H2, there exists a topological group G with discrete normal subgroups K1 and K2 such that H1 is isomorphic to G/K1 and H2 is isomorphic to G/K2.

Group structure on a covering space

Let H be a topological group and let G be a covering space of H. If G and H are both path-connected and locally path-connected, then for any choice of element e* in the fiber over eH, there exists a unique topological group structure on G, with e* as the identity, for which the covering map p : GH is a homomorphism.

The construction is as follows. Let a and b be elements of G and let f and g be paths in G starting at e* and terminating at a and b respectively. Define a path h : IH by h(t) = p(f(t))p(g(t)). By the path-lifting property of covering spaces there is a unique lift of h to G with initial point e*. The product ab is defined as the endpoint of this path. By construction we have p(ab) = p(a)p(b). One must show that this definition is independent of the choice of paths f and g, and also that the group operations are continuous.

Universal covering group

If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a universal cover. By the previous construction the universal cover can be made into a topological group with the covering map a continuous homomorphism. This group is called the universal covering group of H. There is also a more direct construction which we give below.

Let PH be the path group of H. That is, PH is the space of paths in H based at the identity together with the compact-open topology. The product of paths is given by pointwise multiplication, i.e. (fg)(t) = f(t)g(t). This gives PH the structure of a topological group. There is a natural group homomorphism PHH which sends each path to its endpoint. The universal cover of H is given as the quotient of PH by the normal subgroup of null-homotopic loops. The projection PHH descends to the quotient giving the covering map. One can show that the universal cover is simply connected and the kernel is just the fundamental group of H. That is, we have a short exact sequence

1to pi_1(H) to tilde H to H to 1
where tilde H is the universal cover of H. Concretely, the universal covering group of H is the space of homotopy classes of paths in H with pointwise multiplication of paths. The covering map sends each path class to its endpoint.

Lie groups

The above definitions and constructions all apply to the special case of Lie groups. In particular, every covering of a manifold is a manifold, and the covering homomorphism becomes a smooth map. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism.

Two Lie groups are locally isomorphic if and only if the their Lie algebras are isomorphic. This implies that a homomorphism φ : GH of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras

phi_* : mathfrak g to mathfrak h
is an isomorphism.

Since for every Lie algebra mathfrak g there is a unique simply connected Lie group G with Lie algebra mathfrak g, from this follows that the universal convering group of a connected Lie group H is the (unique) simply connected Lie group G having the same Lie algebra as H.

Examples

  • The universal covering group of the circle group T is the additive group of real numbers R with the covering homomorphism given by the exponential function exp: RT. The kernel of the exponential map is isomorphic to Z.
  • For any integer n we have a covering group of the circle by itself TT which sends z to zn. The kernel of this homomorphism is the cyclic group consisting of the nth roots of unity.
  • The rotation group SO(3) has as a universal cover the group SU(2) which is isomorphic to the group of unit quaternions Sp(1). This is a double cover since the kernel has order 2.
  • The unitary group U(n) is covered by the compact group T × SU(n) with the covering homomorphism given by p(z, A) = zA. The universal cover is just R × SU(n).
  • The special orthogonal group SO(n) has a double cover called the spin group Spin(n). For n ≥ 3, the spin group is the universal cover of SO(n).
  • For n ≥ 2, the universal cover of the special linear group SL(n, R) is not a matrix group (i.e. it has no faithful finite-dimensional representations).

References

  • Pontryagin, Lev S. (1986). Topological Groups. 3rd ed., New York: Gordon and Breach Science Publishers. ISBN 2-88124-133-6.
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