Wondering about the isomorphism defined at the bottom to show $\mathbb{R}/\mathbb{Z} \cong \mathbb{R}/2\pi\mathbb{Z}$. The map given is $g(x + \mathbb{Z}) = 2\pi x + 2\pi \mathbb{Z}$ but would it also be correct to define a simpler map, as $g(x) = 2\pi x$. That seems like it would also work and be slightly cleaner to show than the current isomorphism or would that be incorrect?
You're writing down the same map, just by abuse of notation. In particular, the map you are using is taking is sending $x \mapsto 2 \pi x$, and then, in coset representation it is the map the authors use.