Questions tagged [hopf-fibration]
The hopf-fibration tag has no usage guidance.
11
questions with no upvoted or accepted answers
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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-...
5
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297
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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 $(\theta,\,\phi,\,\psi)$ ...
4
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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}...
3
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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,...
2
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313
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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 ...
1
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0
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196
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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/...
1
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0
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64
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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 $\mathbb{S}^n$ ...
1
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307
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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&=...
1
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233
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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 ...
1
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77
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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^...
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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 ...