I do not understand the definition of Hopf fibrations?

Question

How i go about understanding why Hopf fibration is a map,i did not understand the 3-sphere to 2 sphere concept?Could you please explain??

Answer 1

Lemma: If the quaternion $w$ commutates with every pure imaginary, then $w$ is real.If in addition, $w \in \mathbb{S}^{3}$, then $w = \pm 1$

Proposition : There is a continuous surjective homomorphism $\varphi : \mathbb{S}^{3} \longrightarrow \text{SO}(3)$ with $\text{Ker}(\varphi) = \lbrace -1, 1 \rbrace$

Use this to set the map $h : \mathbb{S}^{3} \longrightarrow \mathbb{S}^{2}$ by putting $h(u) = u^{-1}.i.u = \varphi_{u}(i)$.

1) the application h is continuous, surjective and;

2) $h(u) = h(v) \Leftrightarrow u.v^{-1}$ commutes with i, this is, if, and only if, $w = u.v^{-1} = a + bi$ is a common complex number.

It follows that the equivalence relation induced in $\mathbb{S}^{3}$ is the same as that defined $\mathbb{CP}^{1}$ as the quotient space.

By passing the quotient we obtain a homeomorphism $H : \mathbb{CP}^{1} \longrightarrow \mathbb{S}^{2}$. It follows that $H$ fibration is a locally trivial fiber with typical fiber $\mathbb{S}^{1}$

Remark : I do not know if I understood your question, but I hope I have helped.

For more details, see the references:

I) Fundamental Groups and Covering Spaces by Elon Lages Lima (propostion 14, Chapter 3,in my edition )

II) Basic Algebraic Topology. Anant R. Shastri (Hopf Fibration)