In chapter 2 of that book (the chapter is called "Geometry of the monopole spaces"), it is written:

"If we identify gauge-equivalent monopoles (of charge $k$) we obtain a moduli or parameter space which we shall denote by $N_k$. In fact for most purposes it is best to enlarge this space by a circle or phase factor, so getting a space $M_k$. The simplest way to define this is to fix a direction in $\mathbb{R}^3$, (say the $x_1$-direction), use the gauge $A_1 = 0$ and allow only gauge transformations which tend to the identity as $x_1 \to \infty$."

And then they write a bit later

"It follows that $M_k$ is fibered over $N_k$ with fibre $S^1$."

I feel kind of dumb or something, but I am not sure why this is the case.

Here are some thoughts to let you know about my confused state. I get that one can, using a gauge transformation, reduce to the case $A_1 = 0$, by essentially solving some first-order matrix PDE. I think that the residual group of gauge transformations consist of just constant maps into $SU(2)$ (please inform me if I am wrong).

I think that $SO(3)$ acts on $N_k$, by acting as "spatial" rotations in $\mathbb{R}^3$. Somehow, the subgroup of $SO(3)$ which fixes $S^1$ is a copy of $SO(2)$. Is this somehow the circle fiber they are talking about?

Please help clear my confusion if you can. Thank you!