This optional video is very scant on detail because it relies on ideas from topology. It goes without saying that you are welcome to explore these ideas in more depth in a project.
In this video, we will talk about the topology of some groups we've already met, and some new ones.
The group O(n) of matrices M in G L n R such that M transpose M equals the identity is disconnected: it has two components.
If n=2 then O(2) contains the plane rotations around the origin by any angle and the plane reflections in lines passing through the origin. The rotations cos theta, minus sine theta, sine theta, cos theta form a circle as the angle theta varies; the reflections cos theta, sine theta, sine theta, minus cos theta also form a circle as the line of reflection rotates (the angle the line makes with the x-axis is theta over 2).
When we talk about whether a group is simply-connected, we're interested in loops starting and ending at the identity (marked as a dot in the figure). The subgroup of rotations in O(2) contains the identity, and fails to be simply-connected because of loops which wind around the circle.
The two circles are distinguished by the value of the determinant: the rotations have determinant 1 and form a subgroup SO(2) inside O(2) ("special" means determinant 1). The reflections have determinant minus 1 and do not form a subgroup (e.g. they don't contain the identity).
We have the same picture in all dimensions. The orthogonal group O(n) has two components: SO(n) of rotations and O(n) minus SO(n) containing everything with determinant minus 1 (including reflections, but also other things when n is greater than 2). Recall that any orthogonal matrix M has determinant plus or minus 1 because 1 equals det I equals det M transpose M equals det M squared.
Let's focus on SO(n), n greater than or equal to 3. The loop of matrices with cos theta, minus sine theta, sine theta, cos theta in the top left, ones elsewhere along the diagonal, and zeros elsewhere turns out to be non-contractible in SO(n) when n is greater than or equal to 3 (i.e. there is no nullhomotopy), so this is not a simply-connected group. However, if you go twice around this loop, you get something nullhomotopic. That is the matrix with cos 2 theta, sine minus 2 theta, sine 2 theta, cos 2 theta in the top left, ones elsewhere along the diagonal and zeros elsewhere is nullhomotopic (if you're interested in why, look up the Dirac spanner trick).
In fact there are precisely two "homotopy classes" of loops in SO(n) for n greater than or equal to 3 (the fundamental group of SO(3) is Z mod 2). There is another group Spin n and a 2-to-1 homomorphism Spin n to SO(n) such that Spin n is simply-connected. At the level of Lie algebras, little spin n and little s o n are isomorphic but the corresponding groups are not isomorphic. The case n = 3 is discussed in detail in another video and you can do a project about spin groups if you're interested.
More generally, using some ideas from topology, one can prove the following:
For any Lie group G there is a simply-connected Lie group tilde G and a homomorphism F from tilde G to G such that F star from tilde little g to little g is an isomorphism. tilde G is called the universal cover of G.
In particular, Lie's theorem about exponentiating homomorphisms applies to the universal cover of any group. However, it would be nice to give some explicit examples of simply-connected groups. We're going to need to introduct one more non-simply-connected example first...
Define the
The unitary group is not simply-connected: the determinant of a unitary matrix is a unit complex number because 1 equals det of the identity, which equals det of M dagger M, which equals det M conjugate times det M. Therefore we get a map det from U(n) to U(1). The figure below shows a cartoon picture of how U(n) lives over U(1), and indicates a nontrivial loop in U(n) which projects to a nontrivial loop in U(1). An explicit example of such a loop in U(2) is e to the i theta, 0, 0, 1.
The subgroup SU(n) of matrices in U(n) with determinant 1 (the fibre over 1 of det from U(n) to U(1)) turns out to be a simply-connected group (for all n).
First consider n = 2. In another video about SU(2) we see that SU(2) consists of all matrices a,b,minus b bar, a bar such that the absolute value of a squared plus the absolute value of b squared equals 1. We can therefore identify SU(2) with the 3-dimensional sphere S^3, in other words the set of points real part of a, imaginary part of a, real part of b, imaginary part of b in 4-dimensional space such that the sum of their squares equals 1. This is not to be confused with the 2-dimensional sphere S^2 of everyday experience, which is the unit sphere in 3-dimensional space. The number of dimensions is the number of coordinates we need to specify a point on the surface of the sphere, which is 2 (latitude and longitude) for the everyday sphere S^2, but 3 for S^3.
The space S^3 is well-known to be simply-connected. You can prove it using Van Kampen's theorem, but this proof belongs in a topology course.
You can show that SU(n) is simply-connected by induction on n. The sketch proof is as follows. SU(n+1) acts by rotations on C n + 1, so preserves the unit sphere S 2 n + 1 inside C n + 1. We therefore get a map SU(n+1) to S 2 n + 1, M maps to M v for some fixed unit vector v in S 2 n + 1. The preimage of any w in S 2 n + 1, which is the set of M in SU(n+1) such that M v = w, is a copy of SU(n). The map SU(n+1) to S 2 n + 1 is a special kind of map called a fibration with fibres SU(n), and if the fibre (SU(n)) and base (S^{2n+1}) of a fibration are both simply-connected then so is the total space (SU(n+1)). Again, this argument goes way beyond our course, but it is reassuring to know that there are many natural examples of simply-connected Lie groups.
The examples given so far are all compact groups, that is topologically closed groups of matrices whose matrix entries are bounded. There are plenty of noncompact matrix groups, like G L n R (it contains matrices with arbitrarily large entries).
It turns out that any noncompact group deformation-retracts onto a compact subgroup (any so-called "maximal compact subgroup"). This is a fancy way of saying that the noncompact group can be squished down to the compact subgroup without changing the topology, just like how the plane minus a point can be shrunk to a circle (indeed, this is the example of G L 1 C retracting onto U(1)).
For example:
SU(n) inside S L n C is a maximal compact subgroup, so S L n C is simply-connected;
S O n inside S L n R is a maximal compact subgroup, so S L n R is not simply-connected;
U(n) inside G L n C is a maximal compact subgroup, so G L n C is not simply-connected;
O(n) inside G L n R is a maximal compact subgroup so G L n R is not simply-connected.
These results can be proved using the polar decomposition of our matrices. For example, an element of G L 1 C can be written as the 1-by-1 matrix r e to the i theta for some r positive, and the positive real numbers retract onto the subset consisting of 1.