# Spheres in C^2

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