Topological Groups

Topological groups is another rich source of interesting spaces. A topological group if a topological space G together with a continuous map

$$\mu$$ : G x G$$\rTo$$G

satisfying the usual axioms for multiplication in a group (associativity, existence of unit and existence of inverses). Topological groups form a particularly nice family of spaces, as it turns out that the group structure imposes severe restrictions on the topology. There are many natural examples of topological groups. We will mention a few here.

The real and complex numbers. The real and complex numbers are examples of topological fields (and I leave it to you to figure out what that should mean). Thus each one of them has two underlying topological groups; the additive group and the multiplicative group. The additive group of the real numbers R⁺ is a connected topological group. The multiplicative group R^* on the other hand forms a non-connected topological group. Both the additive and the multiplicative groups of the complex numbers are connected.

The Circle and the k-Torus. The circle is a subgroup of the multiplicative group of the complex numbers C^*. Indeed it is the subgroup (as well as the subspace) given by

S¹ = {z $$\in$$ C  |  | z| = 1}.

Notice that S¹ is a subgroup by virtue of the identities | z₁z₂| = | z₁|| z₂| and | z^-1| = | z|^-1. The topology on S¹ is of course the subspace topology. A Cartesian product of k circles - the k dimensional torus

T^k = S¹ x S¹ x ^... x S¹

k times - is also a topological group in a natural way. Namely, the product is given by coordinate wise multiplication

(x₁, x₂,..., x_k) ^. (y₁, y₂,..., y_k) = (x₁y₁, x₂y₂,..., x_ky_k).

The topology on T^k is the product topology or equivalently the subspace topology where T^k is considered as a subspace of C^k.

Groups of Transformations. Let F be a topological field (i.e. a topological space F with two operations a, m : F x F$\rTo$F, called addition and multiplication, satisfying the usual axioms, which are continuous as maps of spaces. Let M_n(F) denote the set of all linear transformations from Fⁿ to itself. A choice of basis enables us to identify M_n(F) with the set of all n x n matrices over F. Thus M_n(F) can be identified with the n²-dimensional vector spaces $\cong$ Fⁿ² and hence becomes a topological space by giving it the product topology induced from the topology of F itself. Furthermore, multiplication and addition of matrices are continuous operations, since they only involve addition and multiplication in F and so they turn M_n(F) into a topological ring.

Exercise 2.6   What is M₁(F)? Can you see immediately that M₁(F) is a topological ring? What about M₂(F)? Try to convince yourself, using the line of argument sketched above that addition of matrices and multiplication of matrices in M₂(F) are continuous operations.

Now consider the subspace GL_n(F) given by all invertible matrices. The topology on GL_n(F) is defined to be the subspace topology. This subspace is not a ring anymore, since addition of matrices does not preserve invertibility, but rather it is a group with respect to matrix multiplication and is called the general linear group of rank n over F. Within GL_n(F) there are certain subgroups of particular importance. One such subgroup is the special linear group SL_n(F) of all matrices of determinant 1. Let us now specialise to specific fields.

Consider first the case F = R. There are subgroups

SO(n) $$\subseteq$$ SL_n(R)      and      O(n) $$\subseteq$$ GL_n(R).

The subgroup O(n) is called the orthogonal group and consists of all transformations which take any orthonormal basis for Rⁿ to another orthonormal basis, or equivalently, the subgroup of all matrices whose columns form an orthonormal basis for Rⁿ. It is an easy exercise in linear algebra to show that the determinant of all matrices in O(n) is $\pm$ 1. The special orthogonal group SO(n) is the subgroup of O(n) of all matrices of determinant 1.

Exercise 2.7   Show that O(1) is isomorphic to the cyclic group of order 2 and that SO(2) is isomorphic as a topological group to the circle group S¹. Conclude that O(2) is homeomorphic as a space to SO(2) x O(1), but that as a group it is not isomorphic to it.

Replacing R by the complex numbers C we get the unitary groups U(n) and the special unitary groups SU(n), which is the subgroup of U(n) of matrices with determinant 1. The determinant of every matrix in U(n) is of absolute value 1 just as before, but in the complex case this means that it is a complex number on the unit circle.

Exercise 2.8   Show that U(1) is isomorphic to the circle group S¹ as a topological group.

It can be shown that SO(n) and SU(n) are all connected spaces. Also U(n) can be shown to be homeomorphic as a space (not a topological group) to the product space SU(n) x S¹ and so it is connected as well.

Exercise 2.9   Show that O(n) has two connected components both homeomorphic to SO(n) for every n.

Another feature of the groups U(n) and O(n) is that they are compact. This also apply to the subgroups SU(n) and SO(n). We closed this preliminary discussion of topological groups by showing that O(n) is compact. The corresponding statement for U(n) is left as an exercise for the reader.

Proposition 2.10   For every n $\geq$ 1 the group O(n) is compact.