In the Hopf fibration, S^{3} is the set of points ^{2} with ^{2} + |W|^{2} = 1

If _{1} + iU_{2}, W = W_{1} + iW_{2}

_{1}W_{1} + U_{2}W_{2}),
2(U_{2}W_{1} - U_{1}W_{2}),
U_{1}^{2} + U_{2}^{2} -
W_{1}^{2} - W_{2}^{2}).

He says in the book that the existence of this map f implies that
the Hopf fibration is a fiber bundle that does not have a cross-section.
That's because if you had a cross-section of the Hopf fibration
on S^{3}, i.e. a map ^{2} --> S^{3} ^{3} --> S^{2} ^{2}, then there would be a continuous
vector field V on S^{2}, where ^{2}. And this would
be a contradiction, since you can't "comb the hair on a sphere". (your
head is not a sphere since your neck is not being combed, gruesome
thought)

So, f has to interpret a point ^{3} as a tangent vector
to a point on S^{2}. You might hope it'd go to a tangent
vector to

_{1}∂/∂U_{2} -
U_{2}∂/∂U_{1} +
W_{1}∂/∂W_{2} -
W_{2}∂/∂W_{1},

is the exact
**kernel** of the natural map from the tangent space of S^{3}
to the tangent space of S^{2}! When you have a map between manifolds,
there is an associated
map between the tangent spaces of the manifolds that is given by
multiplying a tangent vector by the Jacobian matrix of the map. It's the
best linear approximation of the map at a given point.

So, what you
can do *instead* is to see that

_{1}, W_{2}, U_{1}, −U_{2}) =
−p(U_{1}, U_{2}, W_{1}, W_{2}),^{2};

and the vector
_{1}∂/∂U_{2} -
U_{2}∂/∂U_{1} +
W_{1}∂/∂W_{2} -
W_{2}∂/∂W_{1}_{1}, W_{2}, U_{1}, −U_{2}),
so it's a tangent vector at that point too! And it gets sent to the
tangent space at the antipodal point on S^{2} by the natural
map between tangent spaces, and it's **not** in the kernel of the
natural map at that point!

So, set _{1}, W_{2}, U_{1}, −U_{2})
= (one half of) natural map (U_{1}∂/∂U_{2} -
U_{2}∂/∂U_{1} +
W_{1}∂/∂W_{2} -
W_{2}∂/∂W_{1}).

This does all that's wanted.

Since

And as you go around a fiber on S^{3} - a fiber containing
^{iθ},We^{iθ})

I originally solved it, just by looking at what ^{−1}(0,0,1) and
p^{−1}(0,0,−1),

Laura