= Principal circle bundle on real projective three-space
{title2=$\mathbb{RP}^3\longrightarrow\mathbb{CP}^1$}
Write a point of <Real projective space> as a nonzero vector $(w_1,w_2)\in\mathbb C^2$ modulo real nonzero scaling. The map to <Complex projective space> sends its class to the complex line through that vector. A free right action of the <circle group> is $[w]\cdot u=[\lambda w]$ where $\lambda^2=u$; the two possible roots differ by a real sign and therefore give the same projective point. Local smooth square-root branches prove smoothness. Over $[z:1]$, the fibre coordinate is $(w_2/|w_2|)^2$; over $[1:\zeta]$, it is $(w_1/|w_1|)^2$. On the overlap $\zeta=1/z$, the <transition function of a principal bundle> is $u\mapsto(z/\overline z)u$. Its phase winds twice. Ordinary scalar multiplication by $u$ instead has a kernel of order two and is not a <free action of a group>.
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