Topological groups, why need them?

Question

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and topological spaces, but why do we need it? Can anybody help me?

Answer 1

To a topologist, topological groups are interesting in their own right. The group structure actually gives us interesting topological structure, too! One interesting fact is that the fundamental group (an important topological invariant) of a topological group is Abelian, a fact that spectacularly fails to be true in general - any group can be the fundamental group of some space (proof).

To an analyst or a number theorist, certain kinds of topological groups have incredibly useful properties. Locally compact groups are known to have a Haar measure, a measure that (in some sense) respects the group operation (proof; I can't advise reading it, honestly - it's not one of my favorite proofs). A group having a Haar measure means we can import topics from measure theory and essentially do analysis on this group (in particular, Fourier analysis). One important application of this is in algebraic number theory, where the Haar measure lets us do a lot of things we wouldn't be able to do otherwise.

I've heard that the representation theory of topological groups is particularly nice, but I don't have a reference for this that I've read, so take it as you will; but this fact might make them interesting to an algebraist. In particular, one can recover quite a lot of the representation theory of finite groups by working instead with compact groups; there's a powerful proof technique where one 'averages' over the group, and the Haar measure lets one do this for compact groups, too.

The combination of the group structure and the topological structure is very powerful, and makes these objects helpful to mathematicians of all stripes whenever they appear!

Answer 2

The reason that $<$concept$>$ is important is that there are enough examples of $<$concept$>$ to justify making the definition. So I will answer this by giving some examples. The first example that we come across is the real numbers, $(\mathbb{R}, +)$ (by extension all the $\mathbb{R}^n$). We see that the map $sub:\mathbb{R}^2\to \mathbb{R}$ which is defined by $(x,y)\mapsto x-y$ is continuous. Thus $\mathbb{R}$ is a topological group. One example is not enough to justify a definition, so here are a few more:

• All of the matrix groups over $\mathbb{R}$ and $\mathbb{C}$ (and also all of the Lie groups).

• The circle $S^1$, and all the tori, $(S^1)^n$ (these are special cases of Lie groups, but they are important to mention on their own)

• Let $X$ be any (reasonable, see the comment by Mike for a notion of reasonable) space. Then the set of homeomorphisms of the space, which we will denote $\operatorname{Homeo}(X)$ may be topologized (with the compact open topology) and thus is a topological group. (This one is a little more involved than the first two.)

• Any topological vector space, including Banach spaces and Hilbert spaces. These are important because they come up in functional analysis all the time.

Now, you also say that you find the notion of topological groups difficult. Well, let us suppose that we understand spaces and groups separately. Then all these topological groups are spaces that have a group structure such that all of the group structure is continuous. We may slickly define this as a space $X$ such that that we have a group multiplication on $X$, which we will call $*$, such that the map $$\phi:X^2\to X$$ defined by $$(x,y)\mapsto x*y^{-1}$$ is continuous. To help you understand this definition, I would advise you to verify for yourself that the examples above are indeed topological groups (with the possible exception of the third, which I would defer until you know about the compact open topology).

The fact that the group structure is continuous gives our space a property called homogeneity. Roughly, this means that every point "looks" like very other point. To be more precise, every point looks like the identity element. What this really means if we are not being vague is that if we have a small neighborhood of the identity element $U_e$, then, using the continuity of the group operation, we may find a homoeomorphic neighborhood of $g$ for any element $g$, which we will call $U_g$. We define $U_g$ to be $$\{h\in X:g^{-1}h\in U_e\}.$$ A nice exercise is to show the map, $$\theta:U_g\to U_e$$ given by $$g_2\mapsto g*g_2$$ is indeed a homeomorphism.

To see this homogeneity in $\mathbb{R}^2$, let us suppose that we have a small neighborhood of the origin, say $$\{(x,y)\in\mathbb{R}^2: x^2+y^2<\epsilon\}$$ for some $\epsilon$. We also have a small neighborhood of any another point. For example, the neighborhood of $(4,5)$ is given by $$\{(x,y)\in\mathbb{R}^2: (x-4)^2+(y-5)^2<\epsilon\}.$$ Moreover, the map, $$(x,y)\mapsto (x+4, y+5)$$ is the continuous map that takes the first neighborhood to the second.

In conclusion, here are some slides of a talk that might give additional insight, or at least some other new ideas.

Answer 3

Converted from comment to answer: Topological groups are used to study continuous symmetries, and are used very frequently in modern theoretical physics. Local symmetries form the basis for gauge theories - QED is an abelian gauge theory with the symmetry group $U(1)$, quantum chromodynamics, is a gauge theory with the action of the $SU(3)$ group, and the main attraction: the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. As you can see, topological groups are used to study continuous symmetries, and are used very frequently in modern theoretical physics. Indeed, there are many more examples, but an enlarged post be would be more appropriate for Physics. S.E.

Answer 4

Topological groups also arise out of purely algebraic situations. Here are just two of the most common examples.

In number theory one considers the rings of $p$-adic numbers $\mathbb Z_p$ for primes $p$. I won't get into their construction except to note that their topology is built out of purely algebraic considerations, and that $\mathbb Z_p$ is in fact a topological ring, not just a topological group, with field of fractions $\mathbb Q_p$ (a topological field). The topology of these rings/fields greatly simplifies the theory of their algebraic extensions, and we can understand many things about $\mathbb Z$ and $\mathbb Q$ once we understand $\mathbb Z_p$ and $\mathbb Q_p$ for all primes $p$. (We also need to consider $\mathbb R$, which is also a topological field.)

Here's another example. The Galois correspondence between subgroups and intermediate field extensions breaks down when one looks at infinite extensions--there are just too many subgroups to hope for a bijective correspondence. But Galois groups can be expressed as inverse limits of finite groups, which carry the Krull topology. And it turns out that the closed subgroups in this topology correspond to intermediate extensions, with open subgroups corresponding to the finite subextensions (yes, open subgroups are closed).

In the above examples, the topological spaces are a bit weird (they look like Cantor sets--in fact $\mathbb Z_2$ is homemorphic to Cantor's middle thirds set), but their topological structure is exactly what's needed to make sure these groups measure what they are supposed to.