Please list a few topological groups that I should learn about.

Question

I'm going through Munkres' Topology book and there's a lot about topological groups. For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example. Please list some topological groups or rings that are interesting besides the obvious ones like $(\mathbb{C}, +, \cdot)$, and give a reason why they're interesting. Thanks.

Answer 1

1. All of the classical Lie groups. This one should be self-explantatory. This is probably closest to the type of answer you were thinking about.
2. Local fields and their valuation rings (this might be a little advanced). They are a huge part of modern number theory.
3. In general, the matrix ring or the general linear group over a topological ring. This obviously shows up everywhere, from calculus to number theory.
4. Definitely the circle group $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$. This is a topological group that comes up in multiple places in analysis, in particular it's huge in representation theory. It's also important to take a glance at general tori groups $\mathbb{T}^n$.
5. Lattices in real vector spaces. They have trivial topological structure, but seeing how to prove this/exploit this will give you a good sense of how topological group arguments work.
6. Inner product spaces, either finite dimensional or Hilbert spaces. They are a foundational part of modern analysis.
7. As Zhen Lin points out, we should add in profinite groups (inverse limits of finite groups). In particular Galois groups of (for interests sake, let's say infinite) extensions.

Answer 2

The adèles and idèles of a global field are both fundamental to modern number theory. With these topological groups (and their associated Haar measures) it becomes possible to give a direct, unified proof of the finiteness of class number and the Dirichlet unit theorem, though this is just the beginning of their utility. I also suggest looking at the MathOverflow threads What problem do the adeles solve? and Who fixed the topology on ideles?

Answer 3

Also: for a topological space $X$, the group $\mathrm{Homeo}(X)$ (with compact-open topology) and, for a smooth manifold $M$, $\mathrm{Diff}(M)$, with topology of uniform (on compacts) convergence of all partial derivatives with respect to a fixed background Riemannian metric on $M$. Other similar and important groups are the groups of symplectomorphisms and Hamiltonian symplectomorphisms (of symplectic manifolds), contactomorphisms (of contact manifolds) and biholomorphic automorphisms (of complex manifolds).