Questions tagged [hopf-fibration]

The tag has no usage guidance.

11 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
14 votes
0 answers
767 views

How to see the quaternionic hopf map generates the stable 3-stem?

I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example: third-stable-...
Chris Schommer-Pries's user avatar
5 votes
0 answers
297 views

Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
Guest123412341234's user avatar
4 votes
0 answers
788 views

When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?

Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...
Valac's user avatar
  • 615
3 votes
0 answers
85 views

What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post: Is this generalization of the Hopf map for quadratic field extensions surjective? I still would like to know more and, while that post got several votes,...
Malkoun's user avatar
  • 4,981
2 votes
0 answers
313 views

Explicit map with Hopf invariant two in any even dimension

It is known that the Hopf invariant for maps from $\mathbb{S}^{4n - 1} \to \mathbb{S}^{2n}$ is nontrivial (and captures the rational homotopy of the spheres). For $n = 1$, the Hopf fibration provides ...
Jean Van Schaftingen's user avatar
1 vote
0 answers
196 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
Niculae George Razvan's user avatar
1 vote
0 answers
64 views

Largest elementary neighbourhood

The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...
Eduardo Longa's user avatar
1 vote
0 answers
307 views

Two ways to view the three-sphere

Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates \begin{align*} z&= \cos(\theta/2)e^{i(\phi+\psi)}\\ w&=...
sam's user avatar
  • 133
1 vote
0 answers
233 views

A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...
Ali Taghavi's user avatar
1 vote
0 answers
77 views

Example of n-parameter family of real-analytic diffeomorphisms acting on $S^3$, constant on the Hopf fibres

I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image $S^...
user42388's user avatar
  • 143
0 votes
0 answers
59 views

Obstruction to finding a framing for quotient manifolds

The question is rather open-ended but I hope it is concrete enough. If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...
João Lobo Fernandes's user avatar