For questions on Hopf fibrations

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5
votes
1answer
62 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
3
votes
2answers
107 views

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
1
vote
1answer
90 views

Global Section for Hopf Fibration

I want to know the existence of global section of $\pi : M\rightarrow M/G$, where $M$ is a Riemannian manifold with $G$-action. For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no ...
13
votes
1answer
286 views

What is $\pi_{31}(S^2)$?

What is $\pi_{31}(S^2)$ - high homotopy group of the 2-sphere ? This question has a physics motivation: 1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum ...
11
votes
2answers
588 views

What are all these “visualizations” of the 3-sphere?

a 2-sphere is a normal sphere. A 3-sphere is $$ x^2 + y^2 + z^2 + w^2 = 1 $$ My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and ...
7
votes
1answer
394 views

Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange ...