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. A frequently occurring case is a double covering group, a topological double cover in which H has index 2 in G; examples include the spin groups, pin groups, and metaplectic groups.

Roughly explained, saying that for example the metaplectic group Mp2n is a double cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing one element in the symplectic group.

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

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.

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.

Alternatively, the group law on G can be constructed by lifting the group law H × HH to G, using the lifting property of the covering map G × GH × H.

The non-connected case is interesting and is studied in the papers by Taylor and by Brown-Mucuk cited below. Essentially there is an obstruction to the existence of a universal cover which is also a topological group such that the covering map is a morphism: this obstruction lies in the third cohomology group of the group of components of G with coefficients in the fundamental group of G at the identity.

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

This corresponds algebraically to the universal perfect central extension (called "covering group", by analogy) as the maximal element, and a group mod its center as minimal element.

This is particularly important for Lie groups, as these groups are all the (connected) realizations of a particular Lie algebra. For many Lie groups the center is the group of scalar matrices, and thus the group mod its center is the projectivization of the Lie group. These covers are important in studying projective representations of Lie groups, and spin representations lead to the discovery of spin groups: a projective representation of a Lie group need not come from a linear representation of the group, but does come from a linear representation of some covering group, in particular the universal covering group. The finite analog led to the covering group or Schur cover, as discussed above.

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 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