
Spheres in C^2
Regard the 3sphere $\displaystyle S^3 $as a locus of points in $\displaystyle C^2, (x_1+ix_2, x_3+ix_4)$ satisfying $\displaystyle x^2_1+x^2_2 x^2_3+x^2_4=1$. The multiplicative group $\displaystyle U(1)=[z\in{C}$ $\displaystyle z=1]$ acts on $\displaystyle C^2$ by scalar multiplication.
Check that this action restricts to an action on $\displaystyle S^3$ then show that the quotient of $\displaystyle S^3$ by $\displaystyle U(1)$ is homeomorphic to $\displaystyle S^2$.
Show that the preimage of each point in the quotient is homeomorphic to $\displaystyle S^1$ and that the preimages corresponding to any two distinct points in $\displaystyle S^2$ are linked in $\displaystyle S^3$