TITLE: Construction of maps $f:S^3 \to S^2$ with arbitrary Hopf invariant?
QUESTION [8 upvotes]: The well-known Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$ has explicit constructions involving the geometry of $C^2$ and intersections of complex lines with the $3$-sphere.  They don't seem to generalize easily to "higher" Hopf maps from $S^3 \rightarrow S^2$ with Hopf invariant not equal to one.  Are there any simple expressions for those maps?

REPLY [4 votes]: Actually, yes, there is a construction involving complex projective line. 
Consider all points (x1, x2, x3, x4) on a 3-sphere in the 4-dimensional space. Our goal is to map them to $S^2$  which is the same as $CP^1$
To do this, take a quaternion 

$$x_1+x_{2}i+x_3j+x_4k$$

raise it to the $n$-th power (this is that group law on a 3-sphere) and decompose back into two complex numbers $z_1+z_2j$ . Now $z_i:z_i$
is a point of a complex projective line, that's it!