Understanding $S^{2}\times S^{1}$ as a reduction of $SU(2)\times U(1)$ by $U(1)$ I'm trying to undersand the topology of $S^{2}\times S^{1}$ as a result of reduction of $G=SU(2)\times U(1)$ by an $H=U(1)$ subgroup, in other words as: $\frac{SU(2)\times U(1)}{U(1)}$.
I know elements of $G$ can be written as $\left(U,z\right)$ with $U$ an element of $SU(2)$ and $z$ a unit complex number. If we take $a,b\in\mathbb{Z}$ then we have a homomorphism $\phi_{a,b}:U(1)\rightarrow G$ of the form:
$$\phi_{a,b}\left(z\right)=\left(diag\left(z^{b},z^{-b}\right)z^{a}\right)$$ which is injective when $a,b$ are coprime and then the image $H_{a,b}\leq G$. (Any injective homomorphism $\varphi:U(1)\rightarrow G$ is conjugate to some $\varphi_{p,q}$, as its projection to the $SU(2)$-factor may be conjugated to land in the standard maximal torus.)
If I'm understanding this correctly for $a=1$ we simply get $S^{3}$ as our $G/H_{a,b}$ topology (as in electroweak symmetry breaking in the standard model of particle physics). If $b=1$,we get a class of lens spaces (one for each a, the three sphere here being counted as a degenerate lens space)
I'm thinking $S^{2}\times S^{1}$ corresponds to $b=0$. How do I then understand what the different $a$ correspond to (especially in light of $\pi_{1}\left(G/H_{a,b}\right)=\mathbb{Z}/a$ for $a\neq1$)? Note I'm following a paper.
I'm ultimately interested in the connected sum:$$\#_{k}S^{2}\times S^{1}=\#_{k}\frac{SU(2)\times U(1)}{U(1)}$$
and how I would proceed with a similar breakdown hence me wanting to understand what's going on in the trivial case first.

There are some basic isomorphisms that make this much easier to see. First $U(1) \cong S^1$, which you can see be writing out the definition of $U(1)$. Second, $SU(2) \cong S^3$, the $3$-sphere, see wikipedia. Finally there is a standard action of $S^1$ on $S^3$ with quotient $S^2$, called the Hopf fibration. If you represent the $3$-sphere as $SU(2)$ this action is just complex multiplication.
So topologically we have $SU(2) \times U(1) / U(1) \cong S^2 \times S^1$ if the $U(1)$ acts by the Hopf fibration.
