What does $S^1$ do in many branches of math?

Question

There are many isomorphisms of $S^1$: $\hat{\Bbb Z}, \Bbb R/\Bbb Z, U(1), SO(2), SL(1,\Bbb C), \Bbb T^1, \Bbb R\cup \{\infty\},\Bbb R\Bbb P^1 $. Seeing its importance, I'd like to see a synthesis of the roles of $S^1$ in each branch of math behind the isomorphisms. One can start with something like "as an isomorphism with $X$", "$X$ is the only $Y$ with property $z$" and then pursuit the ideas.

Answer 1

Here's a short list off the top of my head in no particular order:

1. It is a dualizing object for locally compact abelian groups (Pontryagin duality), establishing a contravariant autoequivalence $\text{Hom}(-, S^1)$ that restricts to various other well-known contravariant equivalences, e.g. between discrete and compact abelian groups, and between torsion and profinite abelian groups.

2. In algebraic topology, maps out of $S^1$ describe the fundamental group $\pi_1(-)$.

3. Also in algebraic topology, maps into $S^1$ describe the cohomology group $H^1(-, \mathbb{Z})$.

4. In Lie theory, $S^1$ is the simplest compact Lie group (say, connected, and with positive dimension); finite products of copies of $S^1$ describe maximal tori in compact Lie groups.

5. In harmonic analysis, studying its $L^2$ space gives rise to Fourier series of periodic functions (this is also an aspect of Pontryagin duality although very different from the aspect described in 1).